Applied Science, Faculty of
Engineering, School of (Okanagan)
DSpace
UBCO
Sadeghi, Maryam
Doctor of Philosophy - PhD
University of British Columbia
Lattice codes are central to modern communication theory, unifying error correction, efficient transmission, and secure encoding. Among them, Constructions A and D are especially effective in achieving coding and quantization gains over additive white Gaussian noise (AWGN) and block fading (BF) channels. This dissertation examines the design, analysis, and application of algebraic lattice constructions to enhance reliability and performance in these channels. The first part of the work focuses on Construction A lattices built over the ring of imaginary quadratic integers, tailored for AWGN channels. A simplified and rigorous proof is presented for the existence of lattices that are simultaneously good for both coding and mean squared error (MSE) quantization. This construction exploits the arithmetic of imaginary quadratic fields and leverages discrete dithering to streamline the proof technique. The analysis provides foundational insights into the structure and behavior of algebraic lattices, demonstrating their potential for high-performance communication. The second part introduces a novel framework for Construction D lattices suited for block fading channels, where signal quality varies across transmission blocks. For the first time, these lattices are defined using a combination of nested linear codes and prime ideals in number fields. A layered coding framework is adopted, in which a semi-systematic generator matrix is derived to enable structured encoding. The corresponding decoding algorithm combines maximum-likelihood decoding in the early layers with successive cancellation in the deeper layers. This hybrid strategy ensures full diversity and maintains linear complexity with respect to the lattice dimension. Extensive simulations confirm the superior frame error rate (FER) performance of these lattices over their Construction A counterparts, particularly in high-diversity scenarios. The results also reveal that while multi-layer designs offer flexibility, increasing the number of layers may introduce error propagation challenges, highlighting the importance of tuning depth and code rates for practical performance. Altogether, this dissertation presents a unified treatment of algebraic lattice constructions for both AWGN and block fading channels. By combining rigorous theory with practical design and evaluation, the work contributes to the advancement of lattice-based techniques for reliable and efficient communication, with potential applications in coding, modulation, and physical-layer security.
Design and Analysis of Lattice-BasedCommunication Schemes: A Study ofConstructions A and DbyMaryam SadeghiB.Sc., Shiraz University, Iran, 2009M.Sc., AmirKabir University of Technology (Polytechnic), Iran, 2013M.Sc., The University of British Columbia, 2018A DISSERTATION SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinTHE COLLEGE OF GRADUATE STUDIES(Electrical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)September 2025© Maryam Sadeghi, 2025The following individuals certify that they have read, and recommendto the College of Graduate Studies for acceptance, a dissertation entitled:Design and Analysis of Lattice-Based Communication Schemes:A Study of Constructions A and Dsubmitted byMaryam Sadeghi in partial fulfilment of the requirements ofthe degree of Doctor of PhilosophyChen Feng, School of EngineeringSupervisorAnas Chaaban, School of EnineeringSupervisory Committee MemberZheng Liu, School of EngineeringSupervisory Committee MemberUniversity ExaminerShawn Wang, Department of Computer Science, Mathematics, Physics, and StatisticsExternal ExaminerDaniel Panario, School of Mathematics and Statistics- Carleton UniversityiiAbstractLattice codes are central to modern communication theory, unifying er-ror correction, efficient transmission, and secure encoding. Among them,Constructions A and D are especially effective in achieving coding and quan-tization gains over additive white Gaussian noise (AWGN) and block fading(BF) channels. This dissertation examines the design, analysis, and applica-tion of algebraic lattice constructions to enhance reliability and performancein these channels.The first part of the work focuses on Construction A lattices built overthe ring of imaginary quadratic integers, tailored for AWGN channels. Asimplified and rigorous proof is presented for the existence of lattices that aresimultaneously good for both coding and mean squared error (MSE) quan-tization. This construction exploits the arithmetic of imaginary quadraticfields and leverages discrete dithering to streamline the proof technique.The analysis provides foundational insights into the structure and behav-ior of algebraic lattices, demonstrating their potential for high-performancecommunication.The second part introduces a novel framework for Construction D latticessuited for block fading channels, where signal quality varies across transmis-sion blocks. For the first time, these lattices are defined using a combinationof nested linear codes and prime ideals in number fields. A layered codingiiiframework is adopted, in which a semi-systematic generator matrix is de-rived to enable structured encoding. The corresponding decoding algorithmcombines maximum-likelihood decoding in the early layers with successivecancellation in the deeper layers. This hybrid strategy ensures full diversityand maintains linear complexity with respect to the lattice dimension.Extensive simulations confirm the superior frame error rate (FER) per-formance of these lattices over their Construction A counterparts, particu-larly in high-diversity scenarios. The results also reveal that while multi-layer designs offer flexibility, increasing the number of layers may introduceerror propagation challenges, highlighting the importance of tuning depthand code rates for practical performance.Altogether, this dissertation presents a unified treatment of algebraiclattice constructions for both AWGN and block fading channels. By com-bining rigorous theory with practical design and evaluation, the work con-tributes to the advancement of lattice-based techniques for reliable and ef-ficient communication, with potential applications in coding, modulation,and physical-layer security.ivLay SummaryDigital communication is essential in everyday life, enabling video calls,online meetings, messaging, and streaming. Yet, transmitting data overwireless networks can be unreliable due to noise and signal disruptions. Thisresearch explores how mathematical tools called lattice codes can enhancethe reliability and efficiency of digital communication.The first part of the dissertation explores a well-known technique fordesigning codes, called Construction A. Using a simpler mathematical ap-proach, the research shows that these lattices can help send data reliablythrough noisy environments by reducing the chance of errors during trans-mission. In addition to improving reliability, their structure can also supportdata compression, helping to reduce the amount of information that needsto be sent without losing quality.The second part addresses more difficult conditions, such as changingsignal quality over timeâcommon in mobile communication. For this, adifferent type of lattice code, based on Construction D, is studied. Theresearch proposes a new design framework along with an efficient method forinterpreting and recovering the transmitted data, showing that these codescan adapt to challenging conditions while remaining practical for systemswith limited computing power.Overall, this work provides new insights into improving future commu-vnication systems, especially where high reliability is essential.viPrefaceThis dissertation presents research on the construction and analysis ofalgebraic lattices for communication over additive white Gaussian noise(AWGN) and block-fading channels. The main focus is on the developmentof nested lattice codes over imaginary quadratic integer rings and the designof full-diversity lattices based on Construction D for fading environments.The results contained in this dissertation have contributed to severalpublications:â The study on Construction A lattices over imaginary quadratic inte-gers, providing a simpler proof of the existence of good nested latticecodes for AWGN channels, has been published in the IEEE Open Jour-nal of the Communications Society under the title âA Simpler Proof onthe Existence of Good Nested Lattice Codes over Imaginary QuadraticInteger Rings for AWGN Channelâ, co-authored with Renming Qi,Chen Feng, Hassan Khodaiemehr, and Yu-Chih Huang.â The work on full-diversity Construction D lattices for block-fadingchannels has been accepted for presentation at the 2025 IEEE In-ternational Symposium on Information Theory (ISIT), under the titleâDesign and Decoding of Full-Diversity Construction-D Lattices onBlock-Fading Channelsâ, co-authored with Hassan Khodaiemehr andChen Feng.viiâ Further advancements and extensions of these ideas have led to a forth-coming manuscript submitted to the IEEE Transactions on Commu-nications, co-authored with Hassan Khodaiemehr and Chen Feng.The dissertation is organized to reflect the logical development of theresearch. The preliminary chapters introduce the necessary mathematicalbackground, covering modules, algebraic number fields, and lattice theory.The subsequent chapters focus on the construction of algebraic lattices viaConstruction A over imaginary quadratic integer rings and the design andanalysis of Construction D lattices for block-fading channels, including per-formance evaluation in terms of error probability, diversity, and power effi-ciency.Although some parts of the thesis are based on collaborative work, thecontent has been expanded and integrated into a self-contained document,presenting additional proofs, explanations, and results not included in theoriginal publications.I would like to acknowledge the contributions of my co-authors, whosecollaboration has been invaluable throughout the course of this research.Their expertise and insights have greatly enriched the quality of the workpresented in this dissertation.Generative Artificial Intelligence (AI), specifically ChatGPT, was usedonly for grammar, spelling, and writing feedback. It provided support inrefining the technical and written aspects of the dissertation.viiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.1.1 Wireless Communication: Enhancing Reliability andEfficiency . . . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 Cryptography and Cybersecurity: Securing Informa-tion in the Post-Quantum Era . . . . . . . . . . . . . 91.1.3 Lattice-Based Distributed Systems . . . . . . . . . . . 11ix1.2 Background on Digital Communication Systems and ChannelModels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.2.1 The Additive White Gaussian Noise (AWGN) Channel 151.2.2 Block Fading Channels and Their Importance . . . . . 171.2.3 Coding in Fading Channels . . . . . . . . . . . . . . . 181.2.4 Diversity Techniques in Fading Channels . . . . . . . . 191.3 Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 201.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . 23Chapter 2: Preliminaries . . . . . . . . . . . . . . . . . . . . . . 252.1 Modules and Related Structures . . . . . . . . . . . . . . . . 252.2 Algebraic Number Fields . . . . . . . . . . . . . . . . . . . . . 282.2.1 Definition of Number Fields . . . . . . . . . . . . . . . 282.2.2 Rings of Integers in Number Fields . . . . . . . . . . . 292.3 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.3.1 Basic Definitions and Properties . . . . . . . . . . . . 332.3.2 Quantization and Modulo Operations . . . . . . . . . 342.3.3 Covering Radius and Effective Radius . . . . . . . . . 352.3.4 Second Moment and Normalized Second Moment . . . 352.3.5 Lattice Goodness Criteria . . . . . . . . . . . . . . . . 362.3.6 Lattice Point Enumeration Lemmas . . . . . . . . . . 362.3.7 Nested Lattice Codes . . . . . . . . . . . . . . . . . . 372.3.8 Construction A Lattices . . . . . . . . . . . . . . . . . 372.3.9 Construction D Lattices . . . . . . . . . . . . . . . . . 40Chapter 3: Algebraic Construction A over Imaginary QuadraticFields . . . . . . . . . . . . . . . . . . . . . . . . . . . 43x3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2 Construction A over Imaginary Quadratic Integer Rings . . . 453.2.1 Coding Scheme . . . . . . . . . . . . . . . . . . . . . . 473.3 Error Probability Analysis in Lattice Coding . . . . . . . . . 523.3.1 The Generator Matrix is Full Rank with High Proba-bility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.3.2 The Encoding Error Probability . . . . . . . . . . . . 543.3.3 The Decoding Error Probability . . . . . . . . . . . . 583.3.4 The Total Error Probability . . . . . . . . . . . . . . . 653.4 Second Moment and Power Efficiency in Lattice Codes . . . . 663.4.1 Coding Rate . . . . . . . . . . . . . . . . . . . . . . . 683.5 Estimation of Parameters . . . . . . . . . . . . . . . . . . . . 703.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77Chapter 4: Construction D Algebraic Lattices . . . . . . . . . 784.1 Introdution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 784.2 Construction D Lattices over Ring of Integers . . . . . . . . . 824.2.1 Discussion on the Selection of Linear Component Codes 864.2.2 Generator Matrix of Algebraic Construction D Lattices 904.3 Block Fading Channel Model and Performance Metrics . . . . 1024.3.1 Channel Model . . . . . . . . . . . . . . . . . . . . . . 1024.3.2 Error Performance over BF Channels . . . . . . . . . . 1054.3.3 Error Probability and Design Criteria for Lattices overBlock-Fading Channels . . . . . . . . . . . . . . . . . . 1084.3.4 Poltyrev Threshold and Outage Limit in Lattice-CodedBlock-Fading Channels . . . . . . . . . . . . . . . . . . 1114.4 Decoding of Full-Diversity Construction D lattices . . . . . . 113xi4.4.1 Diversity Analysis of the Decoding Algorithm . . . . . 1174.4.2 Complexity Analysis of the Decoding Algorithm . . . 1214.4.3 Parity-Check Enforcement and Its Impact on Diversity 1234.5 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 1254.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Chapter 5: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 138Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142xiiList of TablesTable 4.1 Performance results of full-diversity Construction-Dand Construction-A lattices. . . . . . . . . . . . . . . 135xiiiList of FiguresFigure 3.1 Comparison of n1/t and n1+t for n values of 10, 100,and 1000, showing n1/t dominance at low values of tand rapid growth of n1+t at higher values of t. . . . . 75Figure 4.1 Poltyrev outage limit for full-diversity Construction-D lattices with [N, k2, k1] = [100, 50, 40] and differentdiversity orders. . . . . . . . . . . . . . . . . . . . . . 128Figure 4.2 Decoding of diversity-2 Construction D lattices withm = 2, 7, 10. . . . . . . . . . . . . . . . . . . . . . . . 130Figure 4.3 Decoding of diversity-2 Construction D lattices overQ(â2) for a = 2, 3, 4. . . . . . . . . . . . . . . . . . . 132Figure 4.4 Decoding of diversity-3 Construction D lattices overthe number field Q(ν) with discriminant dK = 148. . 133xivAcknowledgementsI would like to express my deepest gratitude to my supervisor, Prof.Feng, for his invaluable guidance, support, and encouragement throughoutmy research. His expertise, insight, and unwavering belief in my abilitieshave been a constant source of motivation. I am profoundly grateful forthe opportunity to work under his mentorship and for the knowledge he hasimparted to me.I would also like to extend my sincere thanks to Dr. Khodaeimehr forhis continuous support, thoughtful feedback, and mentorship. His contri-butions have been crucial in shaping my work, and I greatly appreciate hisconstructive advice and encouragement.My heartfelt thanks go to my committee members, Prof. Liu and Prof.Chaaban, for their time, effort, and insightful comments during the courseof my thesis. Their feedback has significantly enriched my research, and Iam grateful for their thoughtful suggestions and constructive criticism.I am fortunate to have worked with such a supportive and knowledgeablegroup of scholars, and I deeply appreciate all their efforts in helping mecomplete this research.xvDedicationThis work is lovingly dedicated to my husband, whose unwavering sup-port, patience, and encouragement have been my strength throughout thisjourney. Your belief in me has been my greatest motivation.To my dearest daughter, your smiles and laughter have been my sourceof joy and inspiration, reminding me each day of the importance of perse-verance and love.Thank you both for being my anchor and my light. This accomplishmentis as much yours as it is mine.xviChapter 1IntroductionThe rapid advancement of modern communication and computing tech-nologiesâincluding 5G, the Internet of Things (IoT), and quantum com-putingâhas intensified the need for communication systems that are notonly efficient and reliable, but also secure and scalable. These technologiesoperate in increasingly complex and high-dimensional environments, wheresignal fading, interference, noise, and evolving security threats present sig-nificant challenges. Wireless communication, as a cornerstone of modernconnectivity, underpins critical applications across mobile networks, satel-lite systems, and IoT infrastructures. As demands for faster, more robust,and energy-efficient systems continue to rise, addressing these challengesbecomes imperative. Meeting these demands calls for advanced mathemat-ical tools and coding strategies that can ensure reliability, efficiency, andresilience in next-generation communication systems.Among the most fundamental components of such systems are cod-ing and modulation techniques, which enable accurate data transmissionand efficient use of communication resources. Classical error-correctingcodesâsuch as Reed-Solomon, convolutional, and low-density parity-check(LDPC) codesâhave served this role with great success. These codes in-troduce redundancy to mitigate errors caused by noise and interference.However, in more demanding scenarios such as block-fading channels, mas-1Chapter 1. Introductionsive MIMO systems, or harsh wireless environments, traditional approachesoften face limitations, including increased decoding complexity, reduced per-formance under diversity constraints, and inefficiencies in high-dimensionalsettings. This has motivated the exploration of alternative coding paradigmscapable of meeting the demands of modern systems.Lattice-based coding has emerged as a compelling alternative, partic-ularly suited for high-dimensional and dynamic communication scenarios.Lattices exhibit favorable packing and covering properties, allowing efficientsignal representation while minimizing error rates in noisy or fading environ-ments. Their inherent geometric structure and symmetry lend themselvesnaturally to diverse applications in modulation, coding, and signal shap-ing. Notably, lattice codes have been extensively studied in the context ofadditive white Gaussian noise (AWGN) channels, where they are modeledas infinite, unrestricted codes that achieve high coding gains [44]. It hasbeen shown that such codes can approach the capacity of the AWGN chan-nel, with rates arbitrarily close to capacity while maintaining bounded errorprobability [21, 58]. By selecting appropriate shaping regions, these ideallattices can be converted into practical, capacity-approaching lattice codes.Over the years, several construction techniquesâmost notably Construc-tions A, B, C, D, and Dâ²âhave been proposed to design lattices with tailoredproperties for specific applications [12]. The choice of construction dependson system requirements such as error performance, complexity, and adapt-ability to fading or interference.Construction A, which maps classical block codes into Euclidean space,offers simplicity and compatibility with existing coding techniques. It per-forms well in Gaussian noise environments, balancing error correction capa-bility and implementation ease.2Chapter 1. IntroductionConstruction D, in contrast, employs a multilevel structure that com-bines multiple nested codes to form high-dimensional lattices. This struc-ture enhances diversity and decoding performance, particularly in commu-nication scenarios affected by fading or interference. These include fadingMIMO systems, where multiple antennas are used over time-varying chan-nels, as well as block-fading channels that exhibit frequency or time selec-tivity. Its multistage decoding strategy sequentially recovers the componentcodes, starting from the strongest level, thereby reducing complexity whilemaintaining strong performance. Construction D also supports the creationof lattices with superior packing properties and robust error correction un-der power and bandwidth constraints. While Construction A is suitablefor simpler environments, Construction D is often preferred for scenariosrequiring high reliability and stronger resistance to noise and fading.Beyond the choice of construction, the algebraic structure of latticeshas proven essential in advancing the performance and efficiency of moderncommunication systems. Algebraic lattices, derived from number fields andmodule structures, provide deeper control over the geometric and arithmeticproperties of the lattice. In AWGN channels, they offer improved shapingand coding gains by leveraging number-theoretic techniques to optimize bothstructure and decoding complexity. These constructions enable lattices tooperate closer to theoretical performance limits while maintaining feasibilityfor implementation.Recent research has extended the application of algebraic lattices to morechallenging settings, such as block-fading channels. One notable directioninvolves the use of lattice-based analog joint source-channel coding (JSCC)for low-latency wireless sensor networks (WSNs), where multidimensionallattices encode spatially correlated analog data for transmission over fading31.1. MotivationSIMOmultiple-access channels [55]. This approach leverages the structure ofalgebraic lattices to enhance reliability and reduce latency using decodingstrategies such as MMSE and sphere decoding. Another important con-tribution concerns the construction of well-rounded algebraic lattices fromodd-prime-degree cyclic number fields, including cases where the prime isramified. These constructions demonstrate robust performance and poten-tial applications in secure communication scenarios such as wiretap channels[15]. Parallel developments also explore partition lattices and their con-nection to modular and distributive properties, providing a framework forstructured coding systems and further improvements in fading environments[4].Algebraic lattices thus embody a powerful synergy between lattice the-ory and algebraic number theory, offering a flexible and efficient frameworkfor addressing the complexities of modern wireless communication. Theirunique blend of structure, performance, and scalability positions them askey enablers of future communication systems. This thesis aims to exploretheir potential through theoretical development, practical application, andsimulation-based evaluation. The next section presents the motivation be-hind this study and outlines specific use cases that highlight the importanceand versatility of algebraic lattice codes.1.1 MotivationAlgebraic lattices, deeply rooted in algebraic number theory, have at-tracted considerable interest for their distinctive mathematical propertiesand wide-ranging applications in modern technology. Algebraic lattices offerexceptional error correction capabilities, high packing efficiency, and robust-41.1. Motivationness against noise and interference. These attributes make them invaluablein fields such as wireless communications, cryptography, and secure datatransmission. As demands for secure and efficient information processingcontinue to rise, algebraic lattices present promising solutions to many chal-lenges in these domains. This section outlines the key application areas ofalgebraic lattices, providing compelling motivation for their continued study.1.1.1 Wireless Communication: Enhancing Reliability andEfficiencyThe increasing demands of modern wireless communication systems,ranging from higher data rates to more efficient spectrum usage, necessi-tate the use of advanced mathematical frameworks. Algebraic lattices haveemerged as powerful tools in this context, particularly in systems employingMultiple-Input Multiple-Output (MIMO), Orthogonal Frequency-DivisionMultiplexing (OFDM), and space-time coding. Their inherent structuresupports efficient signal representation and decoding, yielding significantperformance gains in high-dimensional transmission environments.Lattice-Based Modulation for Fading ChannelsFading channels, which introduce signal degradation and distortion, posea fundamental challenge in wireless communication. Algebraic lattices pro-vide an effective solution by enabling full diversity transmission, ensuringrobust and reliable data delivery time-varying channel conditions.Key Advantages: Lattice-based modulation has proven particularly ef-fective in both block-fading and linear fading environments. The geometric51.1. Motivationregularity of lattices allows the design of dense, noise-resilient signal con-stellations. Algebraic lattices, in particular, support space-time codes thatmaintain performance in rapidly changing channels, and their high packingdensity offers inherent robustness against interference and distortion.Lattice Design for Fading Channels: Specialized lattice code con-structionsâsuch as those based on Construction A over totally real numberfieldsâenable full diversity in block-fading scenarios. These lattices inher-ently provide signal space diversity and facilitate efficient decoding. No-tably, recent developments have yielded low-complexity decoding schemeswith linear complexity in lattice dimension, promoting practical deploymentin real-time systems [29].Capacity-Approaching Codes: Lattice coding techniques have demon-strated the potential to approach the Shannon capacity of fading channels. Aunifying framework has been developed that bridges classical lattice theorywith modern wireless applications, achieving near-capacity performance [60].Moreover, lattice-partition-based schemes have shown strong performance innon-orthogonal multiple access (NOMA) settings, outperforming Gaussiansignaling under certain outage constraints [47].Lattice Reduction for Signal DetectionAs MIMO systems scale in complexity, robust and efficient signal de-tection becomes increasingly vital. Lattice reduction (LR) addresses thischallenge by transforming poorly conditioned channel matrices into near-orthogonal equivalents, thereby simplifying detection and improving perfor-mance.61.1. MotivationKey Advantages: By reducing inter-signal correlation, lattice reductionenhances the reliability of signal recovery while maintaining manageablecomputational demands. Techniques like the LenstraâLenstraâLova´sz (LLL)algorithm deliver near-maximum-likelihood (ML) detection accuracy withsignificantly reduced complexity, outperforming traditional detectors suchas zero-forcing (ZF) and minimum mean square error (MMSE) methods.Lattice Reduction-Aided Iterative Detection Algorithm: This ap-proach enhances MIMO-OFDM detection by integrating lattice reductioninto a parallel interference cancellation framework with internal detector it-erations, achieving a 0.8â2 dB performance gain at a bit error rate (BER)of 10â4 [32].Lattice Reduction Precoding: Lattice reduction-based precoding hasbeen applied in underwater optical MIMO systems to improve detectionperformance by enhancing the orthogonality of the channel matrix, therebyfacilitating more reliable signal recovery under challenging propagation con-ditions [65]Dual Shortest-Longest Vector Algorithm: The Dual Shortest-LongestVector (SLV) algorithm presents a low-complexity lattice reduction tech-nique tailored for massive MIMO systems, where it enhances detectionperformance by effectively orthogonalizing the channel matrix, leading toimproved bit error rate outcomes over conventional MMSE-based detec-tors [39].71.1. MotivationPhysical Layer SecurityIn addition to reliability, algebraic lattices contribute to secure wire-less communication at the physical layer. Their algebraic structure enablesefficient wiretap code constructions that inherently protect data againsteavesdropping, particularly in scenarios where adversaries experience infe-rior channel conditions.Lattice Reduction Techniques for Security Incorporating lattice re-duction into secrecy-optimized precoding schemes significantly enhances sig-nal quality for legitimate users while simultaneously impairing the receptioncapabilities of eavesdroppers. This approach strengthens physical layer se-curity in MIMO systems by leveraging the structure of the reduced lattice toincrease decoding complexity for unauthorized receivers. Empirical studiesindicate that lattice reduction-based methods consistently outperform tra-ditional linear precoding techniques in secure communication scenarios [52].Applications in ModernWireless Networks: Lattice-based techniqueshave become foundational in the evolution of modern wireless communi-cation systems, enabling both performance optimization and robust secu-rity across a diverse range of applications. In LTE, 5G, and emerging6G networks, lattice reduction-aided (LR-aided) detection plays a criticalrole in managing the complexity of massive MIMO and multi-user environ-ments, providing reliable signal recovery while maintaining energy efficiencyin dense and resource-constrained settings. Beyond detection, lattices con-tribute to advanced coding strategies such as joint source-channel codingand non-orthogonal multiple access (NOMA), as well as cooperative pro-tocols like beamforming and relaying. Their adaptability extends to op-81.1. Motivationtical communication systemsâincluding underwater and visible light chan-nelsâwhere they address the challenges of power efficiency, interference, andsignal distortion. Importantly, lattice-based physical layer security (PLS) isemerging as a key technology in safeguarding communications in criticalscenarios such as IoT, device-to-device (D2D) networks, satellite links, andUAV communications. These systems benefit not only from the structuralresilience and decoding efficiency of lattices, but also from their inherent re-sistance to quantum adversaries, making them well-suited for post-quantumsecure applications. With their unique combination of high performance,scalability, and cryptographic strength, lattice-based solutions are poisedto be integral components of future-proof, secure, and high-speed wirelessnetworks.1.1.2 Cryptography and Cybersecurity: SecuringInformation in the Post-Quantum EraThe advent of quantum computing presents a serious threat to classi-cal cryptographic systems such as RSA and Elliptic Curve Cryptography(ECC), which rely on mathematical problems that can be efficiently solvedby quantum algorithms like Shorâs. This vulnerability has spurred globalefforts to develop quantum-resistant cryptographic systems, among whichlattice-based cryptography has emerged as a leading contender. Built on thehardness of well-established lattice problems, including the Shortest VectorProblem (SVP) and the Learning With Errors (LWE) problem, lattice-basedcryptographic schemes offer strong security guarantees even against adver-saries equipped with quantum capabilities, positioning them at the forefrontof post-quantum cryptography (PQC).91.1. MotivationKey Advantages: Lattice-based cryptography provides several advan-tages over classical encryption approaches. Its most prominent feature isits conjectured quantum resistance: the underlying lattice problems are be-lieved to remain computationally hard even for quantum computers, al-though this has not been definitively proven. Furthermore, lattice schemessupport a broad range of cryptographic primitives including key encapsu-lation, digital signatures, and advanced functionalities such as fully homo-morphic encryption (FHE), which allows computations on encrypted datawithout the need for decryption. These attributes make lattice cryptographyhighly adaptable for privacy-preserving applications and secure multipartycomputation. The schemes are also scalable and efficient, with relatively lowcomputational overhead, making them suitable for resource-constrained en-vironments such as IoT and large-scale deployments like cloud infrastructureand financial systems.Post-Quantum Cryptography Techniques: Lattice-based cryptogra-phy has been a central focus within the broader field of PQC, with numerousschemes based on the LWE and Ring-LWE problems demonstrating boththeoretical strength and practical viability [38]. In response to emergingthreats, hybrid cryptographic frameworks have been proposed, combiningclassical and post-quantum techniques to ensure compatibility during tran-sitional phases [5]. Such approaches are particularly valuable in mitigatingimmediate quantum risks while providing a path toward long-term quantum-resilient infrastructure.Applications in Modern Cryptographic Systems: Lattice-based cryp-tographic schemes are increasingly being adopted in real-world applications,101.1. Motivationmarking a critical transition from theoretical development to practical de-ployment. One of the most significant milestones in this evolution is theU.S. National Institute of Standards and Technology (NIST) selection of thelattice-based schemes Kyber and Dilithium for future cryptographic stan-dards, designated for key encapsulation and digital signatures, respectively.These protocols are poised to secure a broad range of mission-critical infras-tructures, including financial services, government communication, militaryoperations, and blockchain platforms. Beyond national security and digitalfinance, lattice-based cryptography is gaining traction in cloud computingand Internet of Things (IoT) ecosystems, where it addresses the dual chal-lenge of ensuring strong security while maintaining efficiency in resource-constrained environments. For instance, recent studies have demonstratedthe effectiveness of lattice-based frameworks in secure data aggregation forIoT and cloud systems, offering confidentiality, integrity, and low memoryconsumption without sacrificing computational efficiency [6]. These sys-tems not only outperform conventional aggregation techniques in terms ofperformance and resilience to cyberattacks, but also offer lightweight andscalable solutions suitable for large-scale and embedded deployments. Asthe demand for quantum-resistant and scalable security continues to rise,lattice-based cryptography is emerging as a cornerstone of next-generationcybersecurity infrastructures.1.1.3 Lattice-Based Distributed SystemsLattice-based distributed systems integrate advanced cryptographic tech-niques into decentralized architectures to enhance data security, computa-tional efficiency, and resilience against both classical and quantum threats.111.1. MotivationTraditional distributed systems rely on data partitioning and replicationacross multiple nodes to maintain fault tolerance and high availability; how-ever, they often encounter challenges in safeguarding data integrity, con-fidentiality, and key management. By leveraging hard lattice problemssuch as Learning With Errors (LWE) and Ring-LWE, these systems enablequantum-resistant encryption, robust key exchange, and even homomorphiccomputation, making them ideal for use in cloud storage, edge computing,and blockchain-based networks.Key Advantages: One of the primary advantages of lattice-based ap-proaches in distributed environments is their intrinsic post-quantum secu-rity, ensuring that sensitive data remains protected against emerging quan-tum threats. Cryptographic protocols built on lattice foundations supportsecure secret sharing and access control by distributing encrypted data frag-ments such that only authorized users can reconstruct the full message.Additionally, lattice codes offer powerful error correction capabilities anddata redundancy without excessive storage overhead, thereby enhancingfault tolerance. These schemes also enable secure data deduplicationâawidely used technique in cloud storage and cryptography for eliminating re-dundant dataâwithin multi-tenant systems by allowing verification of du-plicate content without revealing the plaintext. Importantly, lattice-basedsystems facilitate fully homomorphic encryption (FHE), which allows com-putation on encrypted dataâa critical functionality for privacy-preservingapplications in distributed and collaborative environments.Data Distribution Techniques: Recent research has demonstrated theintegration of lattice-based cryptography into efficient data distribution and121.1. Motivationkey management frameworks. For instance, LaKey introduces a distributedpseudorandom function (PRF) scheme that significantly improves the scala-bility of distributed key derivation protocols while maintaining post-quantumsecurity [24]. Moreover, distributed digital signature schemes based on lat-tice constructions enable secure collaborative signing, as explored by Alka-dri et al. [3], offering practical solutions for multi-signer authentication withlow computational overhead. These methods are particularly relevant forhierarchical and peer-to-peer systems where consistency, redundancy, andmodularity are critical.Applications in Modern Distributed Systems: Lattice-based cryp-tography is being actively integrated into a broad spectrum of distributedsystems. In blockchain and decentralized ledgers, lattice-based encryptionsecures transactions and smart contracts against quantum adversaries, asshown in recent infrastructure proposals [25]. In secure multiparty compu-tation and federated learning, fully homomorphic encryption and secure ag-gregation protocols protect data privacy while enabling collaborative modeltraining. IoT and sensor networks benefit from lightweight lattice-based pro-tocols that support secure data aggregation and anomaly detectionâcrucialfor infrastructures like smart grids and industrial automation. Further-more, cloud environments are leveraging forward-secure lattice-based search-able encryption schemes to enable privacy-preserving queries over encrypteddatasets [63]. Biometric authentication systems and identity frameworksalso employ lattice encryption to facilitate secure and private user verifi-cation. Collectively, these applications highlight the pivotal role of lattice-based approaches in driving the next generation of secure, scalable, andquantum-resilient distributed technologies.131.2. Background on Digital Communication Systems and Channel Models1.2 Background on Digital CommunicationSystems and Channel ModelsDigital communication systems form the foundation of modern informa-tion exchange, enabling reliable, high-speed, and efficient data transmissionacross a wide range of technologies, from cellular networks and satellite com-munications to the Internet and Internet of Things (IoT) devices. Thesesystems function by converting discrete source data into signals suitable fortransmission over physical media and then accurately reconstructing thedata at the receiver. Achieving this involves a pipeline of operations thatincludes source encoding, channel encoding, modulation, transmission, de-modulation, and decoding.In the transmission chain, channel encoding introduces structured redun-dancy that allows detection and correction of errors caused by channel im-pairments. Modulation schemes, such as phase shift keying (PSK), quadra-ture amplitude modulation (QAM), and orthogonal frequency-division mul-tiplexing (OFDM), map the encoded bits into waveform representations ap-propriate for the communication channel. At the receiver, demodulationand decoding processes attempt to recover the original message from thenoisy, distorted received signal.The performance of digital communication systems is heavily influencedby the characteristics of the physical channel. While the additive whiteGaussian noise (AWGN) channel model is commonly used to capture ther-mal noise with constant power spectral density, real-world wireless channelsare significantly more complex. In practice, signal transmission is affectedby interference, shadowing, Doppler shifts, and particularly by fading, thephenomenon caused by multipath propagation and movement in the envi-141.2. Background on Digital Communication Systems and Channel Modelsronment.1.2.1 The Additive White Gaussian Noise (AWGN)ChannelThe AWGN channel is a foundational model in digital communicationtheory, representing an idealized scenario where the primary impairmentto transmission is random Gaussian noise. It serves as a benchmark forevaluating the performance of coding and modulation schemes and plays acentral role in information theory due to its mathematical tractability andrelevance to many real-world systems.In this model, the received signal y â Rn is described byy = x+ z,where x â Rn is the transmitted codeword, and z â¼ N (0, Ï2In) is an in-dependent and identically distributed (i.i.d.) Gaussian noise vector withvariance Ï2 per dimension. Here, In denotes the nà n identity matrix.The noise is termed âwhiteâ because its power spectral density is con-stant over all frequencies, and âadditiveâ because it is independent of thetransmitted signal. The simplicity of the AWGNmodel allows for the deriva-tion of analytical performance bounds, most notably Shannonâs capacitytheorem, which defines the maximum achievable data rate C for reliablecommunication as:C =12log2(1 +PÏ2)[bits per channel use],where P is the average transmit power and Ï2 is the noise variance. This151.2. Background on Digital Communication Systems and Channel Modelsexpression represents the theoretical limit on performance under an averagepower constraint.Despite its idealized nature, the AWGN channel provides deep insightinto the behavior of communication systems under Gaussian noise, andits simplicity allows for the development and testing of practical codingschemes.A key advantage of the AWGN channel is its ergodicity, which impliesthat long-term averages over time converge to ensemble averages. This en-ables meaningful use of information-theoretic metrics such as capacity, mu-tual information, and bit error rate (BER) without conditioning on specificchannel realizations.Many classical and modern coding schemes, including convolutional codes,LDPC codes, and turbo codes, have been evaluated and optimized under theAWGNmodel. More recently, lattice codes have emerged as near-optimal forthe AWGN channel in both channel coding and quantization tasks. Latticesare particularly attractive because they:â Allow geometric interpretations of coding gain and decoding regions,â Enable structured, infinite codebooks with good shaping properties,â Support low-complexity decoding methods such as sphere decoding orsuccessive interference cancellation.In the context of this thesis, the AWGN channel serves as a baselinefor evaluating the performance of algebraic lattice constructions. In Chap-ter 3, Construction A lattices over number fields are analyzed under thismodel to demonstrate their error performance, quantization efficiency, andcoding gains. The insights obtained under the AWGN channel inform the161.2. Background on Digital Communication Systems and Channel Modelsdesign and evaluation of these lattices in more complex fading environmentsdiscussed in later chapters.1.2.2 Block Fading Channels and Their ImportanceIn wireless systems, signal fading occurs due to the interaction of elec-tromagnetic waves with environmental objects, resulting in multiple signalpaths from transmitter to receiver. Each path introduces a different delay,attenuation, and phase shift, and the superposition of these componentscauses random fluctuations in the received signal amplitude and phase, com-monly referred to as multipath fading. Depending on the coherence time andmobility conditions, the fading behavior may change over time or frequency.To model this behavior in a tractable manner, block fading channelsare widely used. In these channels, the fading coefficient remains constantover a fixed block of symbols and varies independently from one block tothe next. This reflects real-world conditions in systems using time-divisionor frequency-division multiplexing, where the transmission duration withineach coherence interval is limited. The block fading model is particularlyrelevant in mobile communications, where signal conditions change slowlydue to motion or environmental variations.Block fading channels are non-ergodic, meaning that the channel varia-tions cannot be averaged out over time for each transmitted codeword. Asa result, Shannon capacity is no longer a sufficient metric for system per-formance. Instead, outage probability becomes a central figure of merit. Itquantifies the likelihood that the instantaneous channel conditions are toopoor to support a given transmission rate. A detailed analysis of outagebehavior and its connection to lattice coding is presented in Chapter 4.171.2. Background on Digital Communication Systems and Channel ModelsThe primary challenge in block fading channels is the potential for deepfades that span entire transmission blocks, leading to burst errors that aresignificantly harder to correct than those caused by random noise. Design-ing systems that remain robust under such conditions requires sophisticatedtechniques that can harness redundancy and diversity across multiple chan-nel realizations.1.2.3 Coding in Fading ChannelsRobust coding strategies are essential for ensuring reliable data trans-mission over fading channels, where signal degradation can lead to severeerrors. Classical error-correcting codes, such as Hamming, Reed-Solomon,and convolutional codes, laid the foundation for reliable communication un-der noisy conditions. With the development of turbo codes and low-densityparity-check (LDPC) codes, modern communication systems now approachtheoretical performance limits under many channel conditions, includingfading.More recently, lattice codes have gained attention as an alternative toconventional codes, especially for high-dimensional signal spaces and real-valued communication models. Lattices are naturally suited for channelslike AWGN, and with appropriate construction, they can be extended toperform effectively over fading channels. In particular, algebraic latticeconstructions- such as those derived from number fields via ConstructionA and Construction D- offer modular code design, efficient decoding, andstrong performance guarantees. These algebraic lattices exploit the arith-metic structure of number fields to achieve better packing, higher minimumdistances, and full rank over fading blocks. When extended to block fad-181.2. Background on Digital Communication Systems and Channel Modelsing environments, properly constructed lattice codes can maintain decodingreliability and control error probability even in the presence of deep fades.1.2.4 Diversity Techniques in Fading ChannelsWhile coding introduces redundancy to correct errors, diversity tech-niques are designed to combat fading by exploiting multiple independentchannel realizations. The central idea behind diversity is to reduce theprobability that all copies of a signal experience deep fading simultaneously,thereby improving the robustness of communication.Diversity can be categorized into several forms:â Time diversity: Transmitting coded symbols across different timeinstances to exploit temporal variations in the channel.â Frequency diversity: Spreading the signal across multiple frequencybands or subcarriers (e.g., in OFDM) to reduce sensitivity to frequency-selective fading.â Spatial diversity: Using multiple antennas at the transmitter and/orreceiver (MIMO systems) to create parallel spatial channels that ex-perience independent fading.â Polarization diversity: Transmitting over different signal polariza-tions to increase robustness to propagation changes.In block fading channels, diversity is often achieved by spreading infor-mation across independently fading blocks. One prominent design goal isachieving full diversity, ensuring that the error probability decays exponen-tially with the number of fading blocks. Techniques such as space-time block191.3. Contributioncodes (e.g., Alamouti coding), layered MIMO architectures, and block-wiselattice designs are specifically developed to exploit this structure.This thesis builds on these principles by exploring diversity-achieving al-gebraic lattice constructions and evaluating their error performance in blockfading environments. In particular, we show how the structure of numberfields and their associated prime ideals can be used to construct lattices withguaranteed diversity properties and efficient decoding algorithms.1.3 ContributionThis thesis presents several original contributions to the theory, construc-tion, and application of algebraic lattices in modern communication systems.By combining tools from algebraic number theory with lattice coding tech-niques, this work shows how the arithmetic properties of number fields andtheir rings of integers can be used to design lattices that exhibit full di-versity, strong coding performance, and efficient implementation. Thesecontributions address major challenges in communication theory, includingerror correction, spectral efficiency, and energy optimization, while also cre-ating opportunities for further applications in cryptography and secure dataprocessing.One of the key motivations behind this research is to highlight the advan-tages of algebraic lattices compared to conventional integer lattices. Whiletraditional lattices are defined over the integer ring Zn, algebraic lattices arebuilt from number fields such as imaginary quadratic or cyclotomic fields.These structures enable the construction of lattices with higher packing den-sity, improved minimum distance properties, and more effective error cor-rection. These features make algebraic lattices especially valuable in both201.3. Contributioncoding theory and cryptographic contexts, where their structure supportsfaster computations, compact representations, and stronger security guar-antees based on worst-case to average-case reductions.In communication systems, algebraic lattices also provide efficient spherepacking, which helps improve spectral usage and reduce power consumptionwhile maintaining reliable data transmission. The natural embeddings ofnumber fields into higher-dimensional spaces allow for the construction oftightly packed lattices with enhanced coding efficiency. These propertieslead to measurable gains in performance, reinforcing the value of algebraiclattices in practical wireless communication, secure transmission, and signalprocessing.The main contributions of this thesis are summarized as follows:1. Simplified proof for the existence of good algebraic latticesA simplified and self-contained proof is presented for the existenceof algebraic lattices constructed via Construction A over imaginaryquadratic number fields. These lattices are proven to be suitable forboth coding and quantization in AWGN channels, leveraging the struc-ture of imaginary quadratic integer rings and their prime ideals. Tocomplement the theoretical development, discrete dithering is incor-porated into the coding scheme. This technique simplifies the encoderand decoder design, enhances implementation efficiency, and preserveskey statistical properties such as uniformity and translation invariance.Together, these results offer a practical and mathematically groundedlattice framework for communication systems.211.3. Contribution2. A novel framework for constructing algebraic lattices usingConstruction DA new construction method is introduced for building algebraic latticesover the ring of integers of a number field using Construction D. Thisframework integrates multiple levels of nested linear codes with thealgebraic properties of the number field to create lattices with strongperformance characteristics. One of the key features of this approachis the development of semi-systematic generator matrices, which offeran efficient and organized way to describe these lattices. This designenhances computational practicality and makes the method applicableto high-dimensional and multi-user communication scenarios.3. Performance evaluation of Construction D algebraic latticesin block fading channelsThe proposed lattices based on Construction D are evaluated in blockfading channel environments, which are representative of many real-world wireless communication systems. Through both theoretical anal-ysis and simulation, these lattices are shown to achieve full diver-sity and demonstrate excellent error performance. Their ability tomaintain reliability in the presence of fading and noise makes them apromising solution for advanced wireless networks.To enhance clarity and coherence, a review of the relevant literature isintegrated into each chapter, providing the necessary technical context indirect alignment with the corresponding contributions.221.4. Thesis Organization1.4 Thesis OrganizationThis thesis is organized into five chapters, each addressing a fundamentalaspect of the study and application of algebraic lattices in communicationsystems. The structure is designed to guide the reader from the foundationsof digital communication and lattice theory to the design, analysis, andperformance evaluation of advanced algebraic lattice constructions.Chapter 2 introduces the mathematical preliminaries required for therest of the thesis. It presents the fundamental definitions and propertiesof lattices and algebraic number fields, with a focus on structures relevantto lattice coding. The chapter bridges the gap between abstract algebraictheory and its role in the design of practical communication systems, formingthe theoretical backbone of the work.Chapter 3 examines algebraic lattices constructed using Construction Aover the ring of integers of a number field. It details the construction process,which involves embedding linear codes over finite fields into Euclidean spacevia algebraic number theory. The chapter also analyzes the performance ofthese lattices over additive white Gaussian noise (AWGN) channels, focusingon important metrics such as error probability and quantization efficiency.Chapter 4 is dedicated to the study of Construction D algebraic lattices,which are built using a hierarchy of nested linear codes and prime idealsin number fields. This chapter outlines a new framework for constructingthese lattices, including the derivation of semi-systematic generator matri-ces. The focus is on their behavior in block fading channels, where their fulldiversity and strong error performance are demonstrated through both the-oretical analysis and simulation results. The chapter also discusses decodingstrategies tailored to these channel conditions.231.4. Thesis OrganizationChapter 5 concludes the thesis by summarizing the main findings andcontributions. It reflects on the implications of the research for both theoryand practice, and proposes several directions for future work. These includepotential generalizations of the presented constructions, the exploration ofalternative number fields, and further applications in emerging communica-tion and cryptographic systems.24Chapter 2PreliminariesChapter 2 provides the mathematical foundations required for under-standing the advanced concepts and constructions presented in subsequentchapters. A solid grasp of algebraic number fields, rings and ideals, rings ofintegers, and lattice theory is essential to fully appreciate the structure andproperties of algebraic lattices, as well as their application in communicationsystems.This chapter begins with an overview of algebraic number fields, intro-ducing key definitions and properties. We then explore the theory of ringsand ideals, laying the groundwork for understanding the arithmetic struc-ture of number fields. Next, we examine the ring of integers associated witha number field, which plays a central role in the construction of algebraiclattices. The chapter concludes with an introduction to lattice theory, withparticular emphasis on nested lattice codes and their relevance to the designof structured communication schemes.2.1 Modules and Related StructuresModules generalize vector spaces by allowing scalars to come from a ringrather than a field. This generalization enables the study of linear algebraicstructures in more abstract settings, accommodating both commutative and252.1. Modules and Related Structuresnoncommutative rings. Modules play a central role in algebraic numbertheory and algebraic lattice constructions, particularly when lattices aredefined over rings of integers in number fields.Let R be a ring (not necessarily commutative or with unity, unless statedotherwise). An R-module M is an abelian group (M,+) equipped with anaction of R via scalar multiplication R ÃM â M , denoted (r,m) 7â rm,satisfying the following axioms for all r, s â R and m,n âM :â r(m+ n) = rm+ rn,â (r + s)m = rm+ sm,â (rs)m = r(sm),â If R has a multiplicative identity, then 1R Â·m = m.This structure defines a left module; a right module is defined analo-gously with scalar multiplication acting on the right. In the case of commu-tative rings, the distinction between left and right modules disappears.Several familiar algebraic objects naturally form modules. Every abeliangroup is a Z-module under integer multiplication. For any ring R, the Carte-sian product Rn forms an R-module under component-wise addition andscalar multiplication. Ideals of a ring R are R-submodules of R itself. If Mis an R-module and N â M is a subgroup closed under scalar multiplica-tion, then N is a submodule of M . Quotient modules can also be formed:if N â M is a submodule, the quotient group M/N inherits a natural R-module structure defined by r(m+N) = rm+N .Given twoR-modulesM andN , a map f :M â N is called anR-module262.1. Modules and Related Structureshomomorphism if it preserves both addition and scalar multiplication:f(m1 +m2) = f(m1) + f(m2), and f(rm) = rf(m),for all r â R, m,m1,m2 âM . The kernel of f , denoted ker(f), is a submod-ule of M , and the image im(f) is a submodule of N . If f is bijective, it isan isomorphism, and the modules M and N are said to be isomorphic. Theset of all R-module homomorphisms from M to N , denoted HomR(M,N),forms an abelian group under pointwise addition.A module M is said to be finitely generated if there exists a finite set ofelements {m1, . . . ,mn} âM such that every m âM can be expressed as alinear combinationm = r1m1 + · · Â·+ rnmn, for some ri â R.If such a generating set is also linearly independent and spansM , it is calleda basis, and the module is said to be free.Let R be a ring and M an R-module. For any ideal I â R, one definesIM ={nâk=1akxkâ£â£â£â£â£n â N, ak â I, xk âM}.The subset IM is a submodule of M , and the quotient module M/IMadmits a natural R/I-module structure via the action(r + I)(x+ IM) = rx+ IM,for all r â R and x â M . This action is well-defined and endows M/IMwith the structure of an R/I-module.272.2. Algebraic Number FieldsUnderstanding the behavior of modules under quotienting by ideals isparticularly important in the context of this thesis, especially for the al-gebraic lattice constructions in Chapter 4. The following lemma captureessential properties of module-ideal interactions and will be used in laterconstructions.Lemma 2.1 ([50]). Let R be a ring, I a proper ideal of R, and M a freeR-module with basis B. Then M/IM is a free R/I-module with basis Ï(B),where Ï :M âM/IM is the canonical projection, and |B| = |Ï(B)|.2.2 Algebraic Number FieldsAlgebraic number fields are central objects in algebraic number the-ory. They generalize the field of rational numbers Q by adjoining algebraicelementsâroots of polynomial equations with rational coefficients. Thissection provides formal definitions and properties of number fields, theirembeddings, and their rings of integers, all of which serve as essential toolsfor constructing algebraic lattices in subsequent chapters.2.2.1 Definition of Number FieldsLet K be a subfield of the complex numbers C such that the degree ofthe field extension [K : Q] = n is finite. In this case, K is called a numberfield. The degree n represents the dimension of K as a vector space over Q.An element θ â C is said to be algebraic over Q if it is a root of a non-zero polynomial in Q[x]. If, in addition, θ is a root of a monic polynomialwith coefficients in Z, then θ is called an algebraic integer. The field Q(θ)denotes the smallest subfield of C containing both Q and θ, and is itself anumber field.282.2. Algebraic Number FieldsFor a number field K = Q(θ) of degree n, the minimal polynomial of θhas exactly n distinct complex roots. These correspond to the embeddingsÏ1, . . . , Ïn : K â C, which are injective field homomorphisms fixing Q.These embeddings reflect the complex structure of the field extension.Definition 2.2 ([31]). Let r1 and r2 denote the number of real and com-plex (non-real pairwise conjugate) embeddings of K, respectively. Then thesignature of the number field K is defined as the pair (r1, r2), satisfyingr1 + 2r2 = n.If r2 = 0, the field is said to be totally real; if r1 = 0, the field is totallycomplex, and r2 =n2 .2.2.2 Rings of Integers in Number FieldsIn this section, we recall the basic definitions of monogenic number fields,quadratic number fields, their rings of integers, and the decomposition ofprime ideals, following [31].Let B denote the set of all algebraic integers in C. For a given numberfield K, the ring of integers OK is defined asOK = K â© B,i.e., the set of all algebraic integers contained in K. The ring OK is a freeZ-module of rank n = [K : Q]. Therefore, there exists a Z-basis {Ï1, . . . , Ïn}such that every α â OK can be uniquely written asα =nâi=1ciÏi, with ci â Z.292.2. Algebraic Number FieldsEach embedding Ïi : K â C can be extended to any x â K, producingthe set of conjugates {Ï1(x), . . . , Ïn(x)}. These allow the definition of thenorm and trace of x â K:N(x) =nâi=1Ïi(x), (2.1)Tr(x) =nâi=1Ïi(x). (2.2)If x â OK , then both N(x) and Tr(x) lie in Z. The discriminant of thenumber field K with respect to a Z-basis {w1, . . . , wn} is defined asâK =(det [Ïi(wj)]ni,j=1)2.Definition 2.3 (Monogenic Number Fields). A number field K is calledmonogenic if its ring of integers can be generated by a single algebraic inte-ger. That is, there exists θ â OK such thatOK = Z[θ] ={a0 + a1θ + · · Â·+ anâ1Î¸nâ1â£â£ ai â Z} .In this case, the power basis {1, θ, . . . , Î¸nâ1} serves as a Z-basis for OK .Definition 2.4 (Quadratic Fields). A quadratic field is a number field ofdegree 2 over Q, written as K = Q(âd), where d â Z is square-free. Ifd < 0, the field is imaginary; otherwise, it is real. The ring of integers OKis given by:OK = Z[ξ], where ξ =ï£±ï£´ï£²ï£´ï£³âd, if d â¡ 2, 3 (mod 4),1+âd2 , if d â¡ 1 (mod 4).302.2. Algebraic Number FieldsThe discriminant of K is âK = 4d in the first case and âK = d in thesecond.Definition 2.5 (Prime Ideal Decomposition). Let p â Z denote a rationalprime, that is, a prime element of the ring of integers Z. We use the termârationalâ to emphasize that p lies in Q, distinguishing it from prime ele-ments or prime ideals in algebraic number fields or their respective rings ofintegers. In OK , the ideal pOK may factor into prime ideals:pOK =Lâl=1Pell ,where each Pl is a distinct prime ideal, el is its ramification index, andfl = [OK/Pl : Fp] is its inertia degree. These satisfy the identityLâl=1elfl = [K : Q].If el > 1 for some l, we say that p ramifies in OK . If L > 1, we say p splits;if L = 1, e1 = 1, and f1 = n, then p is said to be inert.Moreover, since all prime ideals in OK are maximal, their residue fieldsOK/P are finite fields of the form Fpf , where f is the inertia degree.Example 2.6. Let K = Q(â2) with OK = Z[â2]. The basis of OK is{1,â2}, and the minimal polynomial of â2 over Q is f(x) = x2 â 2. Theembeddings are Ï1(â2) =â2, Ï2(â2) = ââ2, so:N(â2) = â2, Tr(â2) = 0.The field K is totally real with signature (2, 0), and its discriminant is312.3. LatticesâK = 8.Example 2.7. Let K = Q(ââ3) and OK = Z[(1 +ââ3)/2]. The minimalbasis is {1, Ï}, where Ï = (1 + ââ3)/2. The embeddings are complexconjugates, and we compute:N(ââ3) = 3, Tr(ââ3) = 0.Since r1 = 0 and r2 = 1, the field is totally complex. The discriminant isâK = â3.Lemma 2.8 ([9]). There always exists a natural prime congruent to 1(mod 3) between integers m and 2m where m > 4.Theorem 2.9 ([2]). Let θ â C be a root of the irreducible polynomial x3 +ax+b, where a, b â Z, and define the cubic field K = Q(θ). The discriminantis given byâK = â4a3 â 27b2.If âK is square-free or âK = 4m, where m is square-free with m â¡ 2 or 3mod 4, then {1, θ, θ2} forms an integral basis for OK .2.3 LatticesLattices are fundamental structures that appear in many areas of math-ematics and engineering, such as number theory, cryptography, coding the-ory, and wireless communication. In these fields, lattices are used to solveproblems related to efficient data transmission, signal quantization, and er-ror correction. Their regular and algebraic structure allows for the design322.3. Latticesof reliable and high-performing communication systems, especially in highdimensions.There are several ways to construct lattices, including the well-knownConstructions A, B, C, D, and Dâ². These constructions often build lat-tices from linear error-correcting codes and are chosen based on the desiredproperties, such as packing density, decoding complexity, or resistance tonoise. Among them, Constructions A and D are particularly important forcommunication and lattice-based coding strategies.In this section, we begin by defining basic concepts and properties of lat-tices, including their geometric and algebraic features. We then introduceimportant performance measures such as quantization error and second mo-ment. Finally, we present the theory of nested lattices and describe two im-portant lattice constructionsâConstruction A and Construction Dâwhichare central to this thesis.2.3.1 Basic Definitions and PropertiesA lattice Î â Rn is a discrete subgroup of Rn closed under addition andsubtraction. Equivalently, it can be generated by integer linear combinationsof n linearly independent vectors. That is, ifG â RnÃn is a generator matrixwith linearly independent rows or columns, thenÎ = {Gz | z â Zn}. (2.3)The volume of a lattice, denoted Vol(Î), is defined as the absolute valueof the determinant of its generator matrix:Vol(Î) = |det(G)|. (2.4)332.3. LatticesThis volume represents the size of the fundamental region of the lattice andis used to analyze packing and covering efficiency.2.3.2 Quantization and Modulo OperationsDefinition 2.10 ([1]). A lattice quantizer associated with a lattice Î is afunction QÎ : Rn â Î that satisfies:1. QÎ(x) â Î for all x â Rn,2. QÎ(x+ λ) = QÎ(x) + λ for all λ â Î (periodicity).The most common and optimal quantizer is the nearest-neighbor quantizer,defined asQÎ(x) â argminÎ»âÎâ¥xâ Î»â¥2. (2.5)Here, â¥x â Î»â¥2 denotes the squared Euclidean distance between the inputvector x â Rn and a lattice point λ â Î, and is defined asâ¥xâ Î»â¥2 ânâi=1(xi â λi)2. (2.6)This quantizer maps each input x to the closest lattice point in Î in termsof Euclidean distance.If multiple lattice points minimize the distance, ties may be broken arbi-trarily, provided the tie-breaking rule respects the periodicity condition (2).The quantization error or modulo-lattice operation is defined asx mod Î â xâQÎ(x), (2.7)which lies in the fundamental Voronoi region VÎ of the lattice.342.3. LatticesThe set of all points closest to a given lattice point λ â Î is called itsVoronoi region, denoted VÎ(Î»). The Voronoi region of the origin is writtenVÎ, and forms a fundamental domain of Rn under the action of Î.2.3.3 Covering Radius and Effective RadiusTo analyze how well a lattice covers space, we define the covering radiusrcov(Î) as the smallest radius such that spheres of radius rcov centered atall lattice points cover Rn:rcov(Î) = min {r | Rn â Î +B(0, r)} . (2.8)where B(0, r) denotes the closed Euclidean ball of radius r centered at theorigin, i.e., B(0, r) = {x â Rn | â¥xâ¥ â¤ r}.The effective radius reff(Î) is the radius of a ball whose volume equalsVol(Î):reff(Î) =(Vol(Î)Vn)1/n, (2.9)where Vn is the volume of the unit-radius ball in Rn. Rogers showed thatfor any n, there exists a lattice such that the ratio rcov/reff approaches 1 asnââ [49].2.3.4 Second Moment and Normalized Second MomentThe second moment of a lattice measures the average squared distanceof points in the Voronoi region from the origin:Ï2(Î) =1n Â·Vol(VÎ)â«VÎâ¥xâ¥2 dx. (2.10)The normalized second moment is defined as:352.3. LatticesG(Î) =Ï2(Î)Vol(VÎ)2/n. (2.11)This value plays a key role in quantization efficiency and power-limited com-munication scenarios.2.3.5 Lattice Goodness CriteriaA sequence of lattices {În} is said to be:- Good for MSE quantization iflimnââG(În) =12Ïe, (2.12)which is the minimum possible value.- Good for Poltyrev if, for transmission over the AWGN channel withoutpower constraints, the decoding error probability can be made arbitrarilysmall whenever the noise variance η2 satisfies:η2 <Vol(În)2/n2Ïe. (2.13)2.3.6 Lattice Point Enumeration LemmasTo estimate the number of lattice points within a ball, we use the fol-lowing results.Lemma 2.11 ([42]). Let s â Rn and r > 0. Then,(max{r âân2, 0})nVn â¤ |Zn â©B(s, r)| â¤(r +ân2)nVn.where Vn denotes the volume of the unit ball in Rn, given by Vn = Ïn/2Î(n/2+1) ,362.3. Latticesand Î(Â·) is the Gamma function.Lemma 2.12 ([28]). Let OK be the ring of integers of a number field Kwith discriminant â. Then, for any s â R2n,(max{r â Ï, 0})2n â¤ |OnK â©B(s, r)|µâ¤ (r + Ï)2n, (2.14)where µ = V2n(â|â|/2)n and Ï =â2n|â|2 .2.3.7 Nested Lattice CodesA pair of lattices Îc â Îf is called a nested lattice pair, where Îc is thecoarse lattice and Îf the fine lattice. The set Îf mod Îc forms a nestedlattice code:L(Îc,Îf ) = Îf â© V(Îc). (2.15)This code consists of all points in Îf that lie within the Voronoi region ofÎc. The number of codewords is equal to the volume ratio:|L(Îc,Îf )| = Vol(Îc)Vol(Îf ). (2.16)Nested lattices play a central role in shaping, power control, and multi-usercommunication.2.3.8 Construction A LatticesConstruction A is a classical method that connects linear coding theorywith lattice theory, enabling the systematic construction of lattices fromcodes over finite fields. By embedding discrete codewords into the Euclidean372.3. Latticesspace, it produces structured lattices that inherit the algebraic propertiesof the underlying code. These lattices are particularly useful in a varietyof settings, including efficient sphere packing, low-complexity decoding, andthe design of structured codes for noise-resilient communication.Classical (Integer-Based) Construction ALet p be a prime number, and let C â (Z/pZ)n be a linear code of lengthn. We define the natural reduction map:Ï : Zn â (Z/pZ)n, Ï(x) = x mod p. (2.17)Then, the associated Construction A lattice is given by:Î = {x â Zn | Ï(x) â C}. (2.18)This lattice consists of all integer vectors whose reduction modulo p is acodeword in C. It is naturally endowed with the structure of C and formsa sublattice of Zn.One can also define the sublattice:Îâ² = pZn, (2.19)which leads to a nested lattice pair Îâ² â Î, a structure useful in multi-leveland coset coding.Example 2.13 (Construction A from a Binary Code). Let C â F32 be the[3, 1] binary repetition code defined byC = {000, 111}.382.3. LatticesWe construct a lattice Î â R3 using Construction A with modulus p = 2,by lifting codewords from C into the ambient Euclidean space. Define thenatural component-wise reduction mapÏ : Z3 â F32, Ï(x) = x mod 2.Then the Construction A lattice associated with C is given byÎ = {x â Z3 | Ï(x) â C}.Explicitly, this means Î contains all integer vectors x = (x1, x2, x3) â Z3such that either:(x1, x2, x3) â¡ (0, 0, 0) (mod 2) or (x1, x2, x3) â¡ (1, 1, 1) (mod 2).Hence, the lattice can be expressed as the union of two cosets of thesublattice 2Z3:Î = 2Z3 âª (2Z3 + 1),where 1 = (1, 1, 1). This reflects the fact that the code C has two codewords,and the lattice corresponds to all integer vectors whose mod-2 reductions liein C.This simple example illustrates the fundamental principle behind Con-struction A: lifting the algebraic structure of a linear code into the geometryof Euclidean space to construct lattices.Further generalizations of Construction A using number fields and al-gebraic integers will be developed in the next chapter, where we focus onconstructions over imaginary quadratic fields.392.3. Lattices2.3.9 Construction D LatticesConstruction D is a multilevel method for building lattices from chainsof nested linear codes. Unlike Construction A, which relies on a single linearcode, Construction D is based on a hierarchy of codes across multiple levelsto construct lattices with enhanced minimum distance and coding gain. Thislayered approach makes Construction D particularly effective for applica-tions in multi-level modulation, nested lattice coding, and high-dimensionalsignal processing, especially in block fading and MIMO channels.Definition 2.14. Let p be a prime number, and let C1 â C2 â · · · â Ca â(Z/pZ)n be a chain of nested linear codes, where each Ci is a linear subspaceof (Z/pZ)n with dimension ki, meaning it contains pki codewords. Then theConstruction D lattice Î â Rn is defined as:Î =ï£±ï£²ï£³aâi=1kiâj=1piâ1Î±ijÏË(gj) | αij â {0, 1, . . . , pâ 1}ï£¼ï£½ï£¾+ paZn,where:â {g1, . . . , gn} â (Z/pZ)n is a basis such that the first ki vectors spanCi,â ÏË denotes the natural embedding from (Z/pZ)n to Rn.This construction can be interpreted as a summation of scaled contri-butions from each code level, where the i-th level contributes vectors scaledby piâ1. The addition of paZn ensures that the resulting lattice is full-rankand periodic in Rn.Example 2.15 (Construction D from Nested Binary Codes). Consider two402.3. Latticesnested binary linear codes of length n = 4 over F2, denoted asC1 â C2 â F42,where:â C1 = {0000, 1111} is a [4, 1] repetition code,â C2 = {0000, 1100, 0011, 1111} is a [4, 2] linear code that contains C1.We build a Construction D lattice Î â R4 using a = 2 coding levels andmodulus p = 2. Let the basis vectors {g1, g2} â F42 be such that:â {g1} = {1111} is a basis for C1,â {g1, g2} = {1111, 0011} is a basis for C2,Let ÏË : F42 â {0, 1}4 â R4 be the natural embedding that maps eachcodeword to a real-valued vector.According to the Construction D framework, the resulting lattice is givenby:Î =ï£±ï£²ï£³2âi=1kiâj=12iâ1Î±ijÏË(gj) | αij â {0, 1}ï£¼ï£½ï£¾+ 22Z4,where k1 = 1, k2 = 2, and z â Z4.Each lattice point λ â Î takes the form:λ = α11 · ÏË(1111) + 2 ·(α21 · ÏË(1111) + α22 · ÏË(0011))+ 4z,with αij â {0, 1}. Thus, Î consists of integer vectors in R4 formed by integerlinear combinations of the binary codewords from the nested codes, scaledaccording to their level, and shifted by a coarse lattice 4Z4.412.3. LatticesThis example demonstrates how Construction D systematically buildsa structured lattice by embedding a chain of nested codes into Rn. Themultilevel nature of the construction enables fine-grained control over dis-tance properties and supports applications in high-dimensional coding andmodulation schemes.Construction D is especially useful in scenarios that demand preciseshaping regions, improved error performance, or flexible control over thetrade-off between code rate and diversity. Its adaptability makes it a naturalcandidate for extension to algebraic settings, as explored in the subsequentchapters, where structured algebraic lattices are designed for communicationover block fading channels.42Chapter 3Algebraic Construction Aover Imaginary QuadraticFields3.1 IntroductionThis chapter investigates the design and analysis of nested lattices us-ing Construction A over imaginary quadratic fields, extending the classicalapproach beyond the integer domain. Algebraic lattices derived from ringsof integers in number fields offer strong theoretical guarantees and practicaladvantages in communication systems, particularly for coding over AWGNchannels.Building upon foundational results on lattice coding and quantization[21, 42], this chapter focuses on a class of lattices constructed from lin-ear codes over finite fields, lifted to the ring of integers OK of imaginaryquadratic fields. These lattices derive favorable properties from both theiralgebraic and geometric structures, including dense packing, good minimumdistance, and modularityâmaking them attractive candidates for use in re-liable communication and compute-and-forward schemes.In particular, we examine the framework introduced in [28], which gen-433.1. Introductioneralizes Construction A to arbitrary imaginary quadratic fields and providestheoretical conditions under which the resulting lattices are good for bothchannel coding and mean-squared error (MSE) quantization. Their adaptivescheme, based on limited channel feedback, selects the most suitable ringof integers to optimize compute-and-forward rates. This generalization goesbeyond traditional lattices over Gaussian and Eisenstein integers, offeringgreater design flexibility.Other notable developments in this area include the construction ofnested lattice families over Eisenstein integers, which are specifically op-timized for both quantization and channel coding [57]. In addition, severallattice network coding schemes have been introduced that jointly incorpo-rate shaping techniques and error correction mechanisms [56]. The impactof covering radius on shaping gain and lattice quantization has also beenstudied in detail [37], and random lattice ensembles have been analyzed tobetter understand their packing efficiency [10].Discrete dithering has also emerged as a powerful tool in modern lattice-based schemes, offering improved implementation efficiency and robustnessover continuous dithers. Notably, Qi et al. established that discrete dithersenable the existence of good lattices for coding with high probability [46],and further extensions showed these codes remain effective even in AWGNchannels with erasures [48].In this chapter, we first introduce the algebraic coding scheme for nestedConstruction A lattices over imaginary quadratic rings. We then providea simplified proof of the existence of lattices that are simultaneously goodfor coding and MSE quantization. Our analysis includes estimation of keyparameters of these lattice codes to optimize their performance in practicalscenarios.443.2. Construction A over Imaginary Quadratic Integer Rings3.2 Construction A over Imaginary QuadraticInteger RingsAlgebraic Construction A lattices can be effectively designed over imag-inary quadratic integer rings by utilizing the structure of finite fields arisingfrom ideal factorizations. In particular, the ring of integers OK of a numberfield enables a natural bridge between finite field codes and lattice struc-tures, especially when working within the realm of imaginary quadraticfields. However, when OK is not a principal ideal domain (PID), whichis typically the case for general number fields, additional algebraic care isrequired to construct lattices with desirable structural properties. As dis-cussed in section 2.4, we focus here on imaginary quadratic fields whosering of integers takes the form OK = Z[ξ]. In this chapter, we specificallyconsider rational primes p that split in OK , enabling well-structured latticeconstructions via prime ideal factorizations. That is, we assume that theideal pOK admits a factorization into a product of distinct prime ideals:pOK =mâi=1Pi.We consider a distinguished prime ideal P0 with norm N(P0) = p, ensuringthat the quotient ring Z[ξ]/P0 is isomorphic to the finite field Fp. Thisproperty enables the definition of a reduction map from Z[ξ] to Fp, whichis foundational for lifting linear codes over finite fields to structured latticepoints in Cn or R2n via algebraic embeddings. More concretely, let Ï denotea ring isomorphism between the finite ring Z[ξ]/P0 and the finite field Fp.We define a surjective ring homomorphism Ï : Z[Î¾]â Fp by composing the453.2. Construction A over Imaginary Quadratic Integer Ringscanonical modulo operation with this isomorphism:Ï : Z[ξ] ââ Fp, (3.1)a+ bξ 7ââ Ï((a+ bξ) mod P0). (3.2)This map naturally extends componentwise to vectors, allowing one to re-duce vectors in Zn[ξ] to Fnp . The corresponding inverse lift ÏË : Fnp â Zn[ξ]maps each codeword to a representative vector in the preimage, which iscrucial for constructing lattice codebooks.With this machinery, we can define Construction A lattices over Z[ξ] bylifting linear codes from Fnp . Let C1 and C2 be nested linear codes over Fpwith respective generator matrices G1 â Fk1Ãnp and G2 â Fk2Ãnp , such thatC2 â C1 â Fnp . This nesting implies that G1 can be written in block form asG1 =ï£®ï£°G2Gâ²ï£¹ï£» , (3.3)where both G1 and G2 are full-rank. The algebraic Construction A latticescorresponding to C1 and C2 are then defined as:Î1 = {x â Zn[ξ] | Ï(x) â C1},Î2 = {x â Zn[ξ] | Ï(x) â C2}.(3.4)These sets form Z-modules, and by construction, Î2 â Î1 â Zn[ξ]. Toensure compatibility with power constraints and support signal shaping, we463.2. Construction A over Imaginary Quadratic Integer Ringsintroduce a scaling factor γ > 0, and define the coarse and fine lattices as:Îc = Î³Î2,Îf = Î³Î1.(3.5)The nested structure Îc â Îf â γZn[ξ] forms the basis of nested lat-tice codes, which are instrumental in designing efficient communication sys-tems. These constructions support multi-level encoding, modulo-lattice op-erations, and shaping, making them well-suited for applications in AWGNchannels, compute-and-forward schemes, and network coding. Their alge-braic origin also enables modular arithmetic and decoding techniques thatbenefit from the underlying number-theoretic structure.3.2.1 Coding SchemeThis section presents the proposed coding scheme for nested lattice codesconstructed over imaginary quadratic integer rings, with a particular empha-sis on the integration and role of discrete dithering. Building on the algebraiclattice framework established in the preceding section, we examine how theintroduction of discrete dithering contributes to the robustness, efficiency,and practical viability of lattice-based communication systems. Specifically,dithering is employed to mitigate quantization artifacts and to ensure uni-form signal shaping, thereby improving both theoretical performance andimplementation fidelity.Dithering is a well-established technique in digital communication, usedto decorrelate quantization noise from the source signal. While traditionallattice-based schemes often employ continuous dithers, typically sampledfrom a uniform distribution over the Voronoi region of the coarse lattice,473.2. Construction A over Imaginary Quadratic Integer Ringsthis approach can be computationally demanding and difficult to implementin practical systems. In contrast, discrete dithering, which selects dithervectors from a finite subset of the lattice (e.g., uniformly over Îc â© V(Îc)),offers distinct advantages in both complexity and performance.The discrete nature of the dither aligns naturally with the algebraicstructure of lattices over Z[ξ], enabling efficient integration into digital ar-chitectures. These discrete dither values are easily generated, stored, andreproduced without requiring high-precision hardware or continuous randomnumber generators. As a result, discrete dithering offers a practical alterna-tive that reduces system complexity while preserving essential performancecharacteristics.From a theoretical perspective, discrete dithering plays a crucial role inimproving error performance. By randomizing the position of codewordswithin the shaping region, discrete dithers ensure that quantization errorsare evenly distributed throughout the signal space. This uniformity mit-igates the risk of systematic or correlated errors, which can significantlydegrade decoder performance in the absence of dithering. Additionally, dis-crete dithering supports modulo-lattice operations with lower computationaloverhead, further enhancing system throughput and latency.Moreover, the use of discrete dithers has been shown to facilitate energy-efficient designs. By controlling the distribution of transmitted lattice pointsand reducing the shaping loss, discrete dithering enables operation at loweraverage power while maintaining reliability. This feature is particularly im-portant for power-constrained scenarios such as wireless sensor networks orIoT applications.In summary, discrete dithering is not merely an implementation simplifi-cationâit is a performance-enhancing tool that aligns with the structure of483.2. Construction A over Imaginary Quadratic Integer Ringsalgebraic lattices, improves decoder robustness, and enables energy-efficientencoding. Its integration into the proposed nested lattice coding frame-work is critical to achieving high-rate, low-complexity, and power-efficientcommunication over AWGN channels.EncodingThe encoding process lies at the core of the proposed nested lattice cod-ing framework, which operates over imaginary quadratic integers for trans-mission over AWGN channels. This strategy exploits both the algebraicstructure of the underlying number field and the advantages of discretedithering to enhance error resilience and energy efficiency.Let G1 â Fk1Ãnp be the generator matrix of the fine linear code C1, wherethe entries are drawn independently and uniformly from the finite field Fp.To encode information, a random vector U â Fnp is sampled uniformly, andits discrete dither representation is constructed via the lifting map ÏË : Fnp âZn[Î¾].The lifted dither ÏË(U) introduces controlled randomness that disperseslattice points uniformly across the shaping region, mitigating quantizationnoise and facilitating energy-efficient transmission. The resulting latticecodebook is defined as a collection of shifted cosets of the coarse lattice Îc:C ={ÏË(mGâ²) + ÏË(U) + Îc |m â Fk1âk2p},whereGâ² â F(k1âk2)Ãnp comprises the lower k1âk2 rows ofG1, correspondingto the coset representatives of C1/C2.To transmit a message m â Fk1âk2p , the encoder maps it to the corre-493.2. Construction A over Imaginary Quadratic Integer Ringssponding coset representative in Zn[ξ] and constructs the transmit vector:X =(ÏË(mGâ²) + ÏË(U))mod Îc.There exists a standard isomorphism between Cn and R2n, under whicheach complex vector x â Cn is represented as a real row vector in R2n byconcatenating its real and imaginary parts:Cn â¼= R2n, x = (x1, . . . , xn) 7â (â(x1), . . . ,â(xn),â(x1), . . . ,â(xn)) .This identification allows complex-valued signal models to be equivalentlyanalyzed within a real-valued framework, which is particularly convenientin lattice-based coding and decoding.By construction, X â R2n lies within the Voronoi region V(Îc) of thecoarse lattice, ensuring that the transmitted vector satisfies the power con-straint while maintaining alignment with the lattice geometry. The use ofdiscrete dithering enhances uniformity in the transmitted signal space andsupports better shaping, which in turn improves decoder performance. Ad-ditionally, the compatibility of discrete dithers with the algebraic structureof Z[ξ] simplifies implementation and facilitates low-complexity encoding indigital systems.DecodingThe decoding process is a critical component of the proposed lattice-based coding scheme, where the receiver aims to recover the transmittedmessage vector from the noisy received signal. The decoding procedure relieson the nested lattice structure, along with the discrete dithering mechanism503.2. Construction A over Imaginary Quadratic Integer Ringsintroduced during the encoding process. The process involves several steps,including scaling, dithering removal, and quantization, which collectivelyenable the accurate recovery of the transmitted message.Upon receiving the noisy signal Y = X+Z, where X is the transmittedlattice codeword and Z represents the additive white Gaussian noise, thedecoder first applies a scaling factor α > 0 to the received signal. Thedecoder then searches for a unique message vector mË â Fk1âk2p such that thecorresponding shifted coset ÏË(mËGâ²) + ÏË(U) + Îc minimizes the Euclideandistance to the scaled received signal αY, i.e.mË = argminmd(ÏË(mGâ²) + ÏË(U) + Îc, αY).Here, d(Â·, Â·) denotes the squared Euclidean distance between a coset repre-sentative and the scaled channel output. Specifically,d(t,y) â â¥tâ yâ¥2, (3.6)where â¥ · â¥2 is the squared Euclidean norm.The decoding process proceeds as follows:First, the received signal is scaled by the factor α, yieldingαY = X+W, (3.7)where the effective noise W is given by W = (Î±â 1)X+ αZ. The effectivenoise accounts for both the Gaussian noise and the distortion caused by thescaling factor.Next, the discrete dither vector ÏË(U) is subtracted from the scaled re-513.3. Error Probability Analysis in Lattice Codingceived signal, and the result is reduced modulo the coarse lattice Îc:Î±Y â ÏË(U) (mod Îc). (3.8)The decoder then quantizes the resulting vector using the fine lattice Îf ,yieldingÏË(mËGâ²) = QÎf (αY â ÏË(U)) (mod Îc)= QÎf(Xâ ÏË(U)) (mod Îc) +QÎf (W (mod Îc))= ÏË(mGâ²) +QÎf (W) (mod Îc). (3.9)Finally, the decoder applies the labeling homomorphism Ï to map thelattice vector back to the message space:mËGâ² =mGâ² + Ï(QÎf (W) (mod Îc)). (3.10)The decoding is considered successful if and only if the effective noise quanti-zation satisfies Ï(QÎf (W) (mod Îc)) = 0, which is equivalent to QÎf (W) âÎc. This condition ensures that the quantization error introduced by thenoise does not corrupt the coset representative, allowing for accurate recov-ery of the message vector.3.3 Error Probability Analysis in Lattice CodingIn lattice coding theory for noisy channels, the error probability servesas a critical performance indicator, encompassing failures across all stages ofthe communication pipeline. This includes decoding errors due to channelnoise, as well as structural failures such as the use of rank-deficient generator523.3. Error Probability Analysis in Lattice Codingmatrices and encoding ambiguities arising from improper lattice point selec-tion. Together, these components define the total error probability, whichreflects both algorithmic and algebraic sources of unreliability.This section presents a rigorous analysis of the error probability associ-ated with the proposed lattice coding scheme. Our primary objective is todemonstrate that, under well-defined conditions, the total error probabilityconverges to zero as the lattice dimension n increases. Leveraging the struc-tural properties of nested lattice codes and the effects of design parameterssuch as dither, lattice shaping, and noise statistics, we establish that codingfailures become asymptotically negligible in high-dimensional regimes. Indoing so, we provide formal guarantees of reliability for the proposed con-struction, reinforcing its applicability to scenarios where energy efficiencyand error resilience are of paramount importance.3.3.1 The Generator Matrix is Full Rank with HighProbabilityA fundamental requirement for constructing lattices in coding schemesis that the generator matrix must be full rank to ensure the correct dimen-sionality and structural integrity of the lattice. A full-rank generator matrixguarantees that the lattice spans the entire ambient space without redun-dancy or gap, forming a well-defined structure suitable for error correctionand quantization.The generator matrix G1 fails to be full rank if at least two of its rowsare linearly dependent. LetE1 = {G1 â Fk1Ãnp | rank(G1) < k1} (3.11)533.3. Error Probability Analysis in Lattice Codingrepresent the event where G1 is not full rank. The probability of this event,denoted by Pr(E1), is given byPr(E1) = 1âk1â1âi=0(1â pipn). (3.12)Using this expression, the probability is upper-bounded byPr(E1) â¤ (pk1 â 1)pân < 1pnâk1. (3.13)It is evident that the probability approaches zero as the lattice dimension nincreases, provided that k1 < βn for some 0 < β < 1. This result highlightsthat the likelihood of obtaining a non-full-rank matrix becomes negligible inhigh-dimensional lattices.3.3.2 The Encoding Error ProbabilityEnsuring reliable communication in lattice-based systems requires thateach transmitted coset representative adheres to a predefined energy con-straint, typically enforced through an n dimensional power or shaping regioncentered at the origin. In this setting, the encoding error probability quanti-fies the likelihood that a randomly selected coset representative falls outsidethis region, thereby violating the power constraint and potentially compro-mising system performance.In the following analysis, we derive an upper bound on the encodingerror probability by examining the geometric and probabilistic interactionbetween lattice cosets and the shaping region. This bound provides theo-retical guarantees on the feasibility of the encoding process under practicalpower constraints.543.3. Error Probability Analysis in Lattice CodingThe shaping region B(ânP ) represents the permissible region for thetransmitted signals, where P denotes the average power. The encoding issuccessful if and only if the transmitted lattice point X lies within thisregion:B(ânP ) = {X â R2n | ||X|| â¤ânP}. (3.14)To express the encoding error formally, recall that the transmitted latticecoset is represented asX â ÏË(mGâ²) + ÏË(U) +Îc, (3.15)where ÏË(U) is the discrete dither and Îc denotes the coarse lattice. Theencoding succeeds if and only if the lattice coset intersects with the shapingregion:ÏË(mGâ²) + ÏË(U) +Îc â©B(ânP ) Ì¸= â (3.16)Otherwise, if the intersection is empty, an encoding error occurs.Let the encoding error event be defined byE2(m) = {m | ÏË(mGâ²) + ÏË(U) +Îc â©B(ânP ) = â }. (3.17)Clearly, E2 is equivalent toâlâFk2pI(ÏË(mGâ² + u+ lG2) â B(ânP ))= 0 (3.18)where I is the characteristic function.553.3. Error Probability Analysis in Lattice CodingThe probability of encoding failure is given byPr(E2) = Prï£«ï£¬ï£­âlâFk2pI(ÏË(mGâ² + u+ lG2) â B(ânP ))= 0ï£¶ï£·ï£¸ . (3.19)The lattice coset points are uniformly distributed over the gridγZn[ξ] â© V(γpZn[ξ]), (3.20)which contains pn points. Consequently, the expected value of the charac-teristic function is given byE(I(ÏË(mGâ² + u+ lG2) â B(ânP )))=|γZn[ξ] â©B(ânP )|pn, (3.21)and the variance is expressed as563.3. Error Probability Analysis in Lattice CodingVar(I(ÏË(mGâ² + u+ lG2) â B(ânP )))=|γZn[ξ] â©B(ânP )|pn(1â |γZn[ξ] â©B(ânP )|pn).(3.22)Using Chebyshevâs inequality and pairwise independence of lattice points,the encoding error probability is upper bounded byPr(E2) â¤ pnâk2|Î³Zn[Î¾] â©B(ânP )| â¤(â|â|2)npnâk2(max{0,ânPÎ³ ââ2n|â|2})2nV2n, (3.23)where |â| is the discriminant of the number field and V2n represents thevolume of the 2n-dimensional unit ball in R2n, given byV2n =Ïnn!. (3.24)The second inequality follows from Lemma 2.12 in the preliminaries, whichbounds the number of algebraic integers from OK lying within a Euclideanball.To ensure that the encoding error probability vanishes as n â â, theprime number p and the scaling factor γ must be chosen such thatPr(E2)â 0 as nââ. (3.25)This result highlights the crucial role of lattice geometry, dimension, andpower constraints in minimizing the encoding error probability, paving the573.3. Error Probability Analysis in Lattice Codingway for asymptotically error-free communication.3.3.3 The Decoding Error ProbabilityAs signals traverse communication channels, they are inevitably affectedby noise and other impairments, leading to potential errors in the receivedmessage. In lattice coding, a central design goal is to develop decodingstrategies that mitigate such errors and enable accurate message recovery atthe receiver. This section focuses on analyzing the decoding error probabilityassociated with the proposed lattice coding scheme. By examining the sta-tistical characteristics of the noise and the underlying lattice structure, wederive conditions under which the error probability vanishes asymptoticallyas the lattice dimension increases.Having received the vector Y, the decoding fails if there exists a messagemâ² such thatd(ÏË(mâ²Gâ²) + ÏË(U) + Îc, αY) < d(ÏË(mGâ²) + ÏË(U) + Îc, αY). (3.26)Equivalently, decoding fails precisely when QÎf (W) /â Îc. Let us define theerror event asE3 = {W | QÎf (W) /â Îc}. (3.27)By applying the Total Probability Theorem, there exists a threshold radiusre such that:Pr(E3) â¤ Pr(W /â B(re)) + Pr(QÎf (W) /â Îc |W â B(re)). (3.28)583.3. Error Probability Analysis in Lattice CodingSince the transmitted vector is given byX = ÏË(mGâ²) + ÏË(U) (mod Îc), (3.29)which represents the shortest vector in its respective coset, we derive thefollowing inequality:d(ÏË(mGâ²)+ÏË(U)+Îc, αY) â¤ â¥XâÎ±Yâ¥ = â¥(Î±â1)X+Î±Zâ¥ = â¥Wâ¥. (3.30)For a fixed representative X = x, the norm of the effective noiseW satisfies:â¥Wâ¥2 = (Î±â 1)2â¥xâ¥2 + Î±2â¥Zâ¥2 + 2Î±(Î±â 1)â¥xZT â¥, (3.31)where the inner product term follows the distributionxZT â¼ N (0, â¥xâ¥2Î·2). (3.32)Since it holds with high probability that â¥Xâ¥2 â¤ nP and â¥Zâ¥2 â¤ nη2, andbecauseX and Z are assumed to be nearly orthogonal for sufficiently large n,we approximate the inner product as â¨X,Zâ© â 0. Under these assumptions,we derive an upper bound on the squared norm of W, whereW = (Î±â 1)X+ αZ.Taking the squared Euclidean norm of both sides yields:â¥Wâ¥2 = â¥(Î±â 1)X+ Î±Zâ¥2= (Î±â 1)2â¥Xâ¥2 + Î±2â¥Zâ¥2 + 2Î±(Î±â 1)â¨X,Zâ©. (3.33)593.3. Error Probability Analysis in Lattice CodingNow, using the bounds â¥Xâ¥2 â¤ nP and â¥Zâ¥2 â¤ nη2, and substitutingâ¨X,Zâ© â 0, we obtain:â¥Wâ¥2 â¤ (Î±â 1)2nP + α2nη2. (3.34)To simplify this further, we substitute the optimal value of α, chosen tominimize the upper bound:α =PP + η2. (3.35)Substituting into (3.34) gives:â¥Wâ¥2 â¤ (1 + ϵ) · nPη2P + η2, (3.36)for any arbitrarily small ϵ > 0, reflecting the approximation in the high-probability regime for large n. We therefore define the effective decodingradius re as:re =â(1 + ϵ) · nPη2P + η2. (3.37)Bounding the Probability of W /â B(re)The first step in the analysis is to establish an upper bound for Pr(W /âB(re)). For any ϵ > 0 and α > 0, based on Equation (3.31), the eventW /â B(re) occurs if at least one of the following conditions is met:EX ={X | â¥Xâ¥ >ânP}, (3.38)EZ ={Z | â¥Zâ¥ >ânÎ·2}, (3.39)603.3. Error Probability Analysis in Lattice CodingEO ={XZT | |XZT | > nÎ±ânPÎ·2}. (3.40)Applying the union bound, we obtain:Pr(W /â B(re)) â¤ Pr(EX) + Pr(EZ) + Pr(EO). (3.41)It has been shown that:Pr(â¥Xâ¥ >ânP ) â¤ pnâk2|Î³Zn[Î¾] â©B(ânP )| . (3.42)Since Z â¼ N (0, η2In), Chebyshevâs inequality yields:Pr(â¥Zâ¥ >ânÎ·2) â¤ 1n. (3.43)For EO, we derive:Pr(EO) â¤ Pr(|XZT | > nÎ±ânPÎ·2â£â£ â¥Xâ¥ â¤ ânP)+ Pr(â¥Xâ¥ > ânP ) â¤E(|XZT |2 | â¥Xâ¥ â¤ ânP )n2Î±nPÎ·2+pnâk2|Î³Zn[Î¾] â©B(ânP )| . (3.44)Since for any fixed X = x satisfying â¥xâ¥ â¤ ânP , the inner product xZT â¼N (0, â¥xâ¥2Î·2) follows a Gaussian distribution, we obtain:E(|XZT |2 | â¥Xâ¥ â¤ânP ) â¤ nPη2. (3.45)Therefore, we conclude that:Pr(W /â B(re)) â¤ 1n+ nâ2Î± + 2pnâk2|Î³Zn[Î¾] â©B(ânP )| . (3.46)613.3. Error Probability Analysis in Lattice CodingBounding the Probability of QÎf (W) /â Îc Given W â B(re)Now, we seek an upper bound for Pr(QÎf (W) /â Îc |W â B(re)). Forany fixed coarse lattice Îc, we analyze the probability:Pr(QÎf (W) /â Îcâ£â£W â B(re),G2 = G2) â¤âmâ² Ì¸=mPr(d(ÏË(mâ²Gâ²) + ÏËp(U) + Îc, αY) â¤ â¥Wâ¥ â£â£W â B(re),G2 = G2) â¤âmâ² Ì¸=mPr(d(ÏË(mâ²Gâ²)â ÏË(mGâ²) + Îc,W) â¤ re â£â£W â B(re),G2 = G2).From the distance property:d(ÏË(mâ²Gâ²) + ÏË(U) + Îc, αY)= d(ÏË(mâ²Gâ²) + ÏË(U) + Îc, X+ (Î±â 1)X+ αZ)= d(ÏË(mâ²Gâ²)â ÏË(mGâ²) + Îc, X+W)= d(ÏË(mâ²Gâ²)â ÏË(mGâ²) + Îc, W). (3.47)Since X is uniformly distributed over Î³Zn[Î¾]â©V(Îc) and independent ofGâ², the noise W = (Î±â 1)X+ αZ is conditionally independent of Gâ².Using the Total Probability Theorem:Pr(A â£â£W â B(re),G2 = G2)=â«WâB(re)fËW|WâB(re),G2(W | G2) Pr(A â£â£W â B(re),G2 = G2) dW,where A = d(ÏËp(mâ²Gâ²) â ÏËp(mGâ²) + Îc,W) â¤ re. Additionally, the termfËW|WâB(re),G2(w | G2) refers to the conditional probability density functionofW, restricted to the domain B(re) and normalized such that it integrates623.3. Error Probability Analysis in Lattice Codingto one over that region, conditioned on G2 = G2.Following Loeligerâs approach [34], we obtain the required bound.Since the inequalityd(ÏËp(mâ²Gâ²)â ÏËp(mGâ²) + Îc,W) â¤ reimplies that[ÏËp(mâ²Gâ²)â ÏËp(mGâ²)] (mod Îc) â B(W, re) (mod Îc),it follows thatPr(d(ÏËp(mâ²Gâ²)â ÏËp(mGâ²) + Îc, W) â¤ re â£â£â£G2 = G2)â¤ Pr([ÏËp(mâ²Gâ²)â ÏËp(mGâ²)]mod Îc â[B(W, re)]mod Îcâ£â£â£G2 = G2).On the other hand, the term[ÏËp(mâ²Gâ²)â ÏËp(mGâ²)] (mod Îc)is uniformly distributed over γZn[ξ] â© V(Îc), leading to:Pr([ÏËp(mâ²Gâ²)â ÏËp(mGâ²)]mod Îc â B(W, re) mod Îcâ£â£â£G2 = G2)=â£â£ γZn[ξ] â© V(Îc) â©B(W, re) â£â£pnâk2â¤â£â£ γZn[ξ] â©B(W, re) â£â£pnâk2.633.3. Error Probability Analysis in Lattice CodingTherefore, we obtain:Pr(d(ÏËp(mâ²Gâ²) + ÏËp(U) + Îc, αY) â¤ â¥Wâ¥ â£â£â£W â B(re), G2 = G2)â¤ maxWâB(re)â£â£ γZn[ξ] â©B(W, re) â£â£pnâk2.Similarly, we derive:Pr(QÎf (W) /â Îcâ£â£â£W â B(re), G2 = G2)â¤ pk1âk2 · maxWâB(re)â£â£ Î³Zn[Ï] â©B(W, re) â£â£pnâk2â¤ maxWâB(re)â£â£ γZn[ξ] â©B(W, re) â£â£pnâk1.Consequently, if we define the error event as E3 = {W | QÎf (W) /â Îc},the probability of decoding failure is given by:Pr (E3) â¤ 1n+nâ2Î±+2(â|â|2)npnâk2(max{0,ânPÎ³ â Ï})2nV2n+(reÎ³ + Ï)2nV2n(â|â|2)npnâk1(3.48)where Ï =â2n|â|2.643.3. Error Probability Analysis in Lattice Coding3.3.4 The Total Error ProbabilityBased on Equations (3.13), (3.23), and (3.48), the total error probabilityof the coding scheme is expressed as:Pe = Pr (E1 âª E2 âª E3) â¤ Pr (E1) + Pr (E2) + Pr (E3) â¤1pnâk1+1n+3(â|â|2)npnâk2V2n(max{0,ânPÎ³ââ2n|â|2})2n+(reÎ³+â2n|â|2)2nV2n(â|â|2)npnâk1.(3.49)The total error probability Pe tends to zero as n â â, provided that thefollowing conditions are satisfied:1pnâk1â 0, (3.50)3(â|â|2)npnâk2V2n(max{0,ânPÎ³ââ2n|â|2})2n â 0, (3.51)(reÎ³+â2n|â|2)2nV2n(â|â|2)npnâk1â 0. (3.52)653.4. Second Moment and Power Efficiency in Lattice Codes3.4 Second Moment and Power Efficiency inLattice CodesThe second moment of a lattice is a key parameter in information the-ory and lattice coding, as it captures the average squared Euclidean normof points distributed over a shaping region (typically the Voronoi cell). Thisstatistical measure reflects how dispersed the lattice points are with respectto the origin and plays a decisive role in determining the energy efficiency ofa coding scheme. A smaller second moment implies a tighter concentrationof points around the origin, thereby lowering the average energy per trans-mitted symbolâa highly desirable property in communication systems withpower limitations.Power efficiency becomes especially important in applications such aswireless communication, satellite links, and low-power devices in the Inter-net of Things (IoT). In these settings, minimizing transmission energy with-out sacrificing reliability is a central design goal, and the second momentprovides a direct handle on this tradeoff.Beyond energy considerations, the second moment also impacts criticalsystem metrics such as SNR and decoding error probability. Reducing thesecond moment leads to lower transmission power requirements, which canyield improved SNR for a fixed noise variance, ultimately enhancing robust-ness against channel noise. Moreover, optimizing this parameter contributesto achieving shaping gainâdesigning lattice codes whose point distributionapproximates that of a sphere, thereby approaching the theoretical powerefficiency limit for a given transmission rate.In this section, we establish the existence of a coarse lattice Îc withinthe proposed ensemble whose second moment satisfies the given power con-663.4. Second Moment and Power Efficiency in Lattice Codesstraint. Working in the real-valued space R2n, we normalize the secondmoment accordingly. To bound Ï2(Îc), we analyze the expected squarednorm of a representative lattice point, denoted Î»â, which is assumed to co-incide with or approximate a deep hole in span(Îc) at distance rcov(Îc).Although Î»â is not an average point in the Voronoi region, its energy givesa useful upper bound for codewords selected from our ensemble, especiallywhen rare high-energy points may occur.We use the law of total expectation to partition the contribution of â¥Î»ââ¥2depending on whether it falls inside or outside the power constraint regionB(ânP ). Accordingly, we obtain:Ï2(Îc) â¤ 12nE(â¥Î»ââ¥2)=12n[Pr(Î»â /â B(ânP ))· E(â¥Î»ââ¥2 | Î»â /â B(ânP ))+Pr(Î»â â B(ânP ))· E(â¥Î»ââ¥2 | Î»â â B(ânP ))]â¤ pnâk2|Î³Zn[Î¾] â©B(ânP )| Â·r2cov(Îc)2n+P2.(3.53)The final term P2 arises from the bound on the conditional expectationE(â¥Î»ââ¥2 | Î»â â B(ânP )), which is upper bounded by nP . Since the prob-ability Pr(Î»â â B(ânP )) â¤ 1, we use a loose but sufficient upper bound12n · nP = P2 for this term.To further bound the first term, we apply volumetric estimates on latticepoint counts inside balls in R2n, along with the upper bound on the covering673.4. Second Moment and Power Efficiency in Lattice Codesradius from Lemma 2.12, yielding:Ï2(Îc) â¤(â|â|2)npnâk2Î³2p|â|4V2n(max{0,ânPÎ³ ââ2n|â|2})2n + P2 . (3.54)Here, V2n denotes the volume of the unit ball in R2n.3.4.1 Coding RateThe coding rate of algebraic nested lattices in a 2n-dimensional space isgiven byR =12nlog2(Vol(Îc)Vol(Îf )). (3.55)Assuming spherical shaping regions for both the coarse and fine lattices, weexpress the volume of a lattice as approximately equal to the volume of aball of corresponding radius. Thus, we can write:R =12nlog2(V2nr2ncV2nr2nf)=12nlog2(r2ncr2nf)=12log2(r2cr2f), (3.56)where r2c = nP denotes the shaping radius squared (average power con-straint), and r2f = r2e is the effective decoding radius squared. The decodingradius is defined asre =â(1 + ϵ)nPη2P + η2, (3.57)which accounts for a small decoding margin ϵ > 0 to ensure high-probabilityerror control.683.4. Second Moment and Power Efficiency in Lattice CodesSubstituting this into the rate expression yields:R =12log2ï£«ï£­ nP(1 + ϵ) nPÎ·2P+Î·2ï£¶ï£¸ = 12log2(P + η2(1 + ϵ)η2). (3.58)Defining the signal-to-noise ratio as SNR = Pη2, we further simplify to:R =12log2(1 + SNR)â log2(1 + ϵ). (3.59)This formulation illustrates the dependence of the coding rate on the SNRand the shaping loss parameter ϵ. The term 12 log2(1+SNR) represents thetheoretical upper bound on the achievable rate, while log2(1 + ϵ) quantifiesthe rate loss due to suboptimal lattice shaping. A smaller ϵ corresponds toa more efficient shaping strategy, enabling the coding rate to approach theShannon limit.In the AWGN channel, maximizing the coding rate is essential for en-suring both reliable and efficient data transmission. Lattice-based codingschemes provide a structured approach to achieving near-capacity perfor-mance while imposing practical constraints on power and computationalcomplexity. The utilization of nested lattice structures, combined with op-timized shaping techniques, enables a trade-off between energy efficiencyand error resilience. By designing lattices that minimize ϵ and optimize thesecond moment, communication systems can operate close to the Shannoncapacity of AWGN channels.693.5. Estimation of Parameters3.5 Estimation of ParametersIn this section, we establish the constraints necessary for ensuring thatthe proposed coding scheme is suitable for both error correction and meansquared error (MSE) quantization. Given the upper bounds on total errorprobability and second moment, our objective is to determine the optimalparameter values that minimize these bounds.Following Lemma 2.8, we assume that the prime number p and the scal-ing factor γ are chosen asp = µn1t+ϵ, γ = nâ12tâÏµ, (3.60)for any t, ϵ > 0. Under these assumptions, we obtain the asymptotic behav-iorsÎ³pââ, and Î³2pâ 0. (3.61)These conditions ensure that the system parameters are appropriately scaledto achieve both reliable coding and efficient quantization. By carefully se-lecting t and ϵ, we can balance the trade-off between minimizing the totalerror probability and optimizing the second moment, thereby enhancing theoverall performance of the proposed scheme. Hence, we derive the con-straints under which the following equations are satisfied:1pnâk1â 0, (3.62)(â|â|2)npnâk2(max{0,ânPÎ³ââ2n|â|2})2nV2nâ 0, (3.63)703.5. Estimation of Parameters(reÎ³+â2n|â|2)2nV2n(â|â|2)npnâk1â 0, (3.64)(â|â|2)npnâk2(max{0,ânPÎ³ââ2n|â|2})2nV2n· Î³2p|â|4â 0. (3.65)Since we already assumed k1 < βn for 0 < β < 1, Equation (3.62) goesto zero. In the following, we will define k1 and k2 as functions of n. Weconsiderk1 = nï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£°1 + logpï£«ï£¬ï£¬ï£¬ï£¬ï£¬ï£­(â|â|2)(1â Î´)2nV1n2n(bÎ³â1 +â|â|2)2ï£¶ï£·ï£·ï£·ï£·ï£·ï£¸ï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£»andk2 = nï£®ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯1 + logpï£«ï£¬ï£¬ï£¬ï£¬ï£¬ï£­(â|â|2)(1â Î´)2nV1n2n(aÎ³â1 ââ|â|2)2ï£¶ï£·ï£·ï£·ï£·ï£·ï£¸ï£¹ï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£º.Equations (3.63), (3.64), and (3.65) will be very small as n grows, wherea =âP2and b =â(1 + ϵ)Pη2P + η2are constants. Using the facts thatlimnââ(2n)V1n2n = 2Ïe and r2e < nP , for small ϵ, one can prove that k2 â¤k1 â¤ n.Example 3.1. Parameter Selection for Finite Block LengthTo demonstrate the feasibility of the proposed construction under finite-dimensional settings, we present a numerical example that satisfies the key713.5. Estimation of Parametersdesign constraints derived in equations (3.62)â(3.65). All constructions arecarried out over a totally real cubic number field K = Q(ν), where ν is aroot of the irreducible polynomial f(x) = x3 â x2 â 3x + 1. This field hasdiscriminant |â| = 148, and its ring of integersOK supports the constructionof algebraic lattices used in both quantization and error correction.We consider a block length n = 60, and choose the code alphabet tobe the finite field Fp with prime p = 41. The associated linear codes arenested, with dimensions k2 = 16 and k1 = 18, satisfying k2 < k1 < n.These parameters are selected to meet the trade-off between quantizationperformance and error correction reliability.The lattice scaling factor is set to γ â 0.050, which governs the powerof the shaped lattice codewords. The constants used for bounding the errorprobability and the second moment are defined asa =âP2=â0.5 â 0.707, b = 0.04, P = 1, η2 = 0.01, δ = 0.01.The volume of the 2n-dimensional unit ball is approximated using theasymptotic identity limnââ(2n)V1/n2n = 2Ïe â 17.079, yielding V 1/n2n â0.142 for n = 60. This allows us to compute the relevant bounds andconfirm that:â The rate condition 1pnâk1 â 0 is satisfied.â The shaping condition in equation (3.63) is met with the chosen k2.â The overall error exponent in equation (3.65) remains exponentiallysmall with increasing n.723.5. Estimation of ParametersHence, the selected parametersn = 60, p = 41, k1 = 18, k2 = 16, γ â 0.050serve as a valid configuration demonstrating the practical applicability ofthe proposed scheme under finite-dimensional conditions. These values en-sure that both error correction and quantization objectives are achievedsimultaneously.Example 3.2. Assume K = Q[ââ3] with discriminant â = â3 andOK = Z[Ï] where Ï = 1 +ââ32. For any integer n, the volume of the lattice(Z[Ï])n is(â32)n. Let P be a prime ideal in the factorization of pOK withnorm N(P) = p. Let Îc and Îf be a nested lattices generated according toSection 3.2. Hence, for t > 0 and δ â (0, 1), by assigning parameters p =µn1t+ϵ, γ = nâ12tâÏµ, k1 = nï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£°1 + log3ï£«ï£¬ï£¬ï£¬ï£¬ï£¬ï£­(â32)(1â Î´)2nV1n2n(â(1 + ϵ)Pη2P + η2n12t+ϵ +â32)2ï£¶ï£·ï£·ï£·ï£·ï£·ï£¸ï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£»,and k2 = nï£®ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯1 + log3ï£«ï£¬ï£¬ï£¬ï£¬ï£¬ï£­(â32)(1â Î´)2nV1n2n(âP2n12t+Ïµ ââ32)2ï£¶ï£·ï£·ï£·ï£·ï£·ï£¸ï£¹ï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£º, Equations (3.62)-(3.65)are satisfied for any ϵ > 0.Alternatively, in the context of Construction A lattices, the achievablecoding rate R can be characterized algebraically by the dimensions of theunderlying nested linear codes. Specifically, if the fine and coarse latticesare constructed from linear codes C1 and C2 â C1 over the finite field Fp,with respective dimensions k1 and k2, then the index of the lattice nesting733.5. Estimation of Parametersis given by|Îc/Îf | = pk1âk2 .Substituting this into the rate expression yields:R = limnââk1 â k22nlog2 p= limnââ12log2ï£«ï£¬ï£¬ï£¬ï£¬ï£¬ï£­âP2 ââ22 Î³(â(1 + ϵ)Pη2P + Î·2+â32Î³)2ï£¶ï£·ï£·ï£·ï£·ï£·ï£¸=12log2(P + Î·2Î·2)â log2(1 + ϵ). (3.66)In contrast, according to [46], assume we have scaled integer lattices Îc andÎf such that Î³Îc â Î³Îf â Î³(â32Z)nwith the same volume of the latticesdefined previously. Therefore, p = µn1+t and γ = nâ12t for any t > 0,k1 =ân(1â logp((rent2â 12 +â34)â(1â Î´)nV2nn))âandk2 =ân(1â logp((âPnt2 ââ34)â(1â Î´)nV2nn))â.It is evident that in the case of quadratic lattice, by setting p = µn1t+ϵ,we get a slower-growing prime number that ensures a denser lattice pack-ing. To see this more clearly, we compare the behavior of the functions n1/tand n1+t for different values of n and t in Figure 3.1. At lower values oft, n1/t is significantly higher than n1+t. As t increases, n1/t decreases and743.5. Estimation of Parameters0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0t102104106108Logarithm of Function ValueComparison of n1/t and n1+ t for Different Values of nn1/t for n = 10n1+ t for n = 10n1/t for n = 100n1+ t for n = 100n1/t for n = 1000n1+ t for n = 1000Figure 3.1: Comparison of n1/t and n1+t for n values of 10, 100, and 1000,showing n1/t dominance at low values of t and rapid growth of n1+t at highervalues of t.n1+t increases, eventually leading to a crossover point where n1+t becomeslarger than n1/t. For larger n, the crossover point occurs at a smaller t.This behavior is crucial for achieving higher data rates because it allows fordenser packing within a given volume. The higher density of lattice pointsdirectly translates into increased throughput, which is essential for high-capacity communication channels. Additionally, this choice of p facilitatesefficient use of the available bandwidth, making it well-suited for moderncommunication systems that demand high spectral efficiency. Furthermore,the scaling factor γ = nâ1/(2t) plays a pivotal role in maintaining a balancebetween error performance and data rate. By adopting this scaling factor,we achieve a more conservative decrease in the minimum distance betweenlattice points as the dimension n increases. This gradual reduction in dis-753.5. Estimation of Parameterstance helps preserve the robustness of the lattice against noise, ensuringreliable communication even in challenging environments. This is partic-ularly valuable for applications requiring high reliability and low bit errorrates.763.6. Conclusion3.6 ConclusionIn this chapter, we thoroughly explored the application of ConstructionA lattices over imaginary quadratic integers, focusing on their effectivenessfor AWGN channels. We demonstrated how algebraic lattices, constructedusing Construction A, offer significant advantages in coding and quantizationcontexts when compared to traditional integer lattices.A key aspect of our approach was the use of discrete dither, which playeda pivotal role in simplifying the proofs related to the existence of latticesthat are well-suited for coding. By employing this technique, we were able tostreamline the process and provide clearer insights into the lattice properties.Our analysis confirmed that, under certain constraints, the error proba-bility of the proposed coding scheme diminishes as the dimension increases,illustrating the robustness of the coding scheme. Furthermore, we observedthat the second moment of the coarse latticeâdirectly tied to the averagepower of transmissionâremains small, indicating its suitability for effectivequantization. The parameters we identified for our imaginary quadraticlattices outperformed those previously established for integer lattices, vali-dating the superior performance and efficiency of our approach.Overall, the results presented in this chapter underscore the benefits ofusing imaginary quadratic integers in lattice construction and offer promis-ing advancements in the field of lattice-based coding for communicationchannels.77Chapter 4Construction D AlgebraicLattices4.1 IntrodutionConstruction D is a fundamental technique for generating lattices fromhierarchically nested linear codes, first introduced by Barnes and Sloane [7].By structuring lattices across multiple levels, this method enables precisecontrol over key lattice parameters, making it especially valuable in codingtheory, cryptography, and wireless communications. A notable theoreticalresult shows that Construction D lattices, along with other multilevel cosetcodes, can approach the sphere packing bound [23]. This highlights thevolume-to-noise ratio (VNR) as a central performance metric, linking latticetheory with information-theoretic optimality.However, realizing these theoretical benefits in practice poses significantchallenges. The primary design goal is to construct structured lattice config-urations that not only approach performance limits but also permit efficientdecoding. Ideally, such lattices should achieve vanishing error probabilitiesas the VNR approaches unity, thereby reaching capacity. Yet, the computa-tional complexity of maximum-likelihood (ML) decoding in high dimensionsoften limits practical deployment.784.1. IntrodutionTo mitigate this, low-complexity decoding strategies have been exten-sively explored. Soft-decision multistage decoding, in particular, has proveneffective across several lattice constructions, including Construction A, D,and Dâ² [13, 16â19, 30, 53, 64]. These advances have been fueled by diversechoices of nested linear codes, such as turbo and BCH codes. For exam-ple, Matsumine et al. proposed a refined multistage decoder tailored toBCH-based Construction D lattices [36]. Their ordered statistics decoderachieved state-of-the-art performance in 128-dimensional lattices, outper-forming turbo-based and Construction Dâ² counterparts. Similarly, polar-based Construction D lattices developed by Yan and Ling demonstratedsuperior coding gains compared to Reed-Muller lattice structures [64].While decoding challenges in AWGN channels have been extensively ad-dressed, adapting lattice-based schemes to BF channels introduces a newlayer of complexity [43]. BF channels, often encountered in frequency-hopping and OFDM systems, are non-ergodic in nature, resulting in zeroShannon capacity under classical coding frameworks [8]. The capacity-achieving potential of lattice codes in ergodic fading point-to-point andmultiple-access channels (MAC) has been established by Hindy and Nos-ratinia [26]. They also proposed an efficient decoding method for ergodicMAC settings, which achieves rates close to capacity â within a constantgap at moderate-to-high SNRs and with a gap that diminishes quadraticallyat low SNRs.To address this, algebraic lattices, derived from the ring of inte-gers of number fields, have emerged as a compelling solution. Their inherentfull-diversity properties make them naturally suited for communication overBF channels [41], particularly for broadcast fading (BF) channels, owingto their full-diversity properties [27, 28, 35, 61]. Additionally, it has beendemonstrated in recent research that algebraic lattice codes, when properly794.1. Introdutionconstructed, can achieve the capacity of compound BF channels [11]. Byharnessing the multiplicative structure of number fields, these codes facili-tate uniformly reliable decoding across varying channel states using a single,fixed lattice decoder.A key contribution of this work is the introduction of a semi-systematicgenerator matrix, which facilitates efficient encoding and decoding whilepreserving fundamental algebraic properties. This structured design ensurespractical implementation feasibility while maintaining strong coding charac-teristics. To validate the effectiveness of these lattices, their performance isanalyzed in block-fading environments, demonstrating advantages in errorprobability reduction and diversity-multiplexing trade-offs. The motivationfor employing algebraic lattices in block-fading channels is rooted in theirability to provide reliable data transmission under varying channel condi-tions. A formal definition of Construction D algebraic lattices is established,highlighting their structural properties and theoretical foundations. Theirkey performance characteristics are examined and compared with algebraiclattices derived from Construction A, emphasizing improvements in codingefficiency, and decoding complexity. Finally, their applicability to block-fading channels is analyzed through both theoretical insights and numericalsimulations, validating their superior performance over existing lattice con-structions.Beyond their effectiveness in communications, Construction D latticeshave demonstrated strong applicability in various fields such as cryptog-raphy and physical-layer security. Compared to Construction A lattices,Construction D lattices have been shown to provide better performance andlower decoding complexity [22]. Feng et al. analyzed the properties ofConstruction D lattices and demonstrated that their nominal coding gain804.1. Introdutionsurpasses that of Construction A lattices. Moreover, the lower decodingcomplexity of Construction D lattices makes them more practical for high-dimensional applications. Several studies have explored different extensionsof Construction D lattices. In [51], the authors proposed turbo latticesusing Construction D and analyzed their properties. Similarly, Yan andLing showed polar-based Construction D lattices outperform Reed-Mullerlattices in terms of coding gain [64]. In their recent contribution, Liu andLing [33] have shown that multilevel Construction-D lattices, constructedover number fields like Q(â2), can attain full diversity and closely approachthe ergodic capacity in fast fading environments, all while ensuring low-complexity decoding through the application of nested polar codes. Despitethese developments, only limited research has been conducted on algebraicextensions of Construction D lattices.A notable study by Wang et al. introduced new lattice constructions (ÏAand ÏD) over finite rings [62]. Their work primarily focused on ConstructionÏA, a generalization of Construction A, and demonstrated its effectivenessfor channel coding under multistage decoding in AWGN channels. Whiletheir approach was limited to principal ideal domains (PIDs), the approachtaken in this work generalizes Construction D lattices over the full ring of in-tegers of a number field. This broader framework enables enhanced diversityand improved performance across a wider range of fading environments. Byextending Construction D lattices into algebraic settings, particularly overthe ring of integers OK of a number fieldK, additional structural advantagesare introduced, enhancing their utility across multiple domains. Previousresearch has explored how the classical Construction D framework, as wellas Forneyâs code formula, can be extended from binary codes to linear codesover finite rings Zq with q > 2 [54]. One significant generalization involves a814.2. Construction D Lattices over Ring of Integersnovel zero-one addition operation in Zq, which aligns with the Schur productfor binary codes. It has been established that these extended constructionsresult in valid lattices if and only if the underlying nested codes remainclosed under this operation.These findings are particularly valuable given the increasing role of q-arylattices in modern cryptographic applications. More recent studies have an-alyzed the structural properties of lattices obtained through ConstructionsD and Dâ², particularly in the context of q-ary linear codes. These investiga-tions have examined their connections to Construction A, derived explicitgenerator matrices, formulated volume expressions, and established theo-retical bounds on their minimum distances [20]. Through this study, Con-struction D algebraic lattices are established as a promising alternative forreliable and efficient communication in fading environments, demonstratingsignificant improvements in error resilience, coding efficiency, and diversityperformance.4.2 Construction D Lattices over Ring of IntegersIn this section, we introduce a new algebraic construction, referred to asConstruction D, based on the arithmetic of monogenic number fields and thestructure of their prime ideal decompositions. This construction defines asequence of ring homomorphisms linked to powers of a principal prime ideal,offering a novel way to encode information into layered module structuresover OK . While inspired by classical constructions in coding and latticetheory, this framework is tailored to the context of algebraic number theoryand represents a new direction in the study of structured reductions overnumber fields.824.2. Construction D Lattices over Ring of IntegersLet θ be an algebraic integer. Consider the number field K = Q(θ), andsuppose its ring of integers is given by OK = Z[θ]. While this structure doesnot hold universally for all number fields, it applies to a special subclassknown as monogenic number fields, for which there exists an element thatsimultaneously generates both the field K and its ring of integers OK [29].Let the ideal generated by p in OK factor aspOK =Lât=1Pett ,where each Pt is a distinct prime ideal in OK and et is the ramificationindex for t = 1, . . . , L. Assume that among these, there exists an index j0such that Pj0 is a principal ideal, generated by some element αj0 â OK .Consequently, for any integer i â¥ 1, we havePij0 = â¨Î±ij0â©.We further assume that the algebraic norm of Pj0 is equal to p, i.e.,N(Pj0) = p,where the algebraic norm N(P) of a prime ideal P â OK is defined asN(P) â |OK/P|,that is, the cardinality of the residue field OK/P. This assumption impliesthat the inertia degree fj0 of Pj0 is equal to 1.834.2. Construction D Lattices over Ring of IntegersUnder this assumption, the extension of residue fields satisfies[OKPj0: Fp]= 1.For each i â {1, . . . , a}, we define a ring homomorphismÏi : OK ââPiâ1j0Pij0(4.1)by the rule:Ïi(x) =[xâ(x mod Piâ1j0)]mod Pij0 . (4.2)To verify that Ïi is well-defined, observe that for any x â OK , the elementxË = x â(x mod Piâ1j0)lies in Piâ1j0 . Since Pj0 is principal, it follows thatxË = Î±iâ1j0 r for some r â OK , and hence Ïi(x) â Piâ1j0 /Pij0 , as desired.Let R = OK , and set both the module and the ideal in Lemma 2.1 to beM = I = Pj0 = αj0OK . Under this choice, we observe that the product IMcorresponds to P2j0 . As a result, the quotient Pj0/P2j0inherits the structureof an OK/Pj0-module.Since we have assumed that Pj0 has norm p, and inertia degree fj0 = 1,it follows that OK/Pj0 â¼= Fp. Thus, Pj0/P2j0 becomes a vector space overFp. We now define the module homomorphismÏ : OK/αj0OK ââ αj0OK/α2j0OK (4.3)by the ruleÏ(r + αj0OK) = αj0r + α2j0OK . (4.4)It is straightforward to verify that Ï is an isomorphism of OK/αj0OK-844.2. Construction D Lattices over Ring of Integersmodules. As a result, we obtain the chain of isomorphismsOK/Pj0 â¼= Pj0/P2j0 â¼= Fp, (4.5)where each is interpreted as a vector space over Fp.This relationship naturally extends by induction: for each i â¥ 1, settingM = αij0OK , R = OK , and I = αj0OK , we conclude that the quotientPiâ1j0 /Pij0â¼= Fp (4.6)as Fp-vector spaces, for all i â N.Let C1 â C2 â · · · â Ca be a chain of nested linear codes over Fp, whereeach code Ci has block length N and dimension ki, i.e., Ci â FNp is an[N, ki]-code for i = 1, . . . , a.By choosing a suitable basis {m1, . . . ,mN} of FNp , we ensure that foreach i, the first ki basis vectors span Ci. In particular, the generator matrixMi â FkiÃNp of Ci can be taken asCi ={x = wiMiâ£â£â£wi â Fkip } , for i = 1, . . . , a. (4.7)We now proceed to define the Construction D lattice over OK , using thelayered ideal structure and the previously defined homomorphisms Ïi. LetΦi : ONK ââ(Piâ1j0Pij0)N(4.8)denote the component-wise extension of Ïi in (4.2), given explicitly byΦi(x) = (Ïi(x1), . . . , Ïi(xN )) , for all x = (x1, . . . , xN ) â ONK . (4.9)854.2. Construction D Lattices over Ring of IntegersSimilarly, we define Ψi to be the component-wise extension of the isomor-phism Ï introduced earlier, mapping from(Piâ1j0 /Pij0)Ninto (Fp)N , usingthe natural vector space isomorphism:Ψi :(Piâ1j0Pij0)Nââ FNp . (4.10)With this notation in place, we define the Construction D lattice Î â ONKasÎ ={x â ONKâ£â£Î¨i (Φi(x)) â Ci for all i = 1, . . . , a} . (4.11)This definition encodes the layered structure of the nested codes Ci into theideal-theoretic framework of OK , resulting in a rich lattice structure thatreflects both the arithmetic of the number field and the hierarchy of codesover Fp.4.2.1 Discussion on the Selection of Linear ComponentCodesThe selection of linear component codes in Construction-D lattice de-sign is influenced by several practical and theoretical considerations. First,it is important to consider how the lattice decoding strategy interacts withthe decoding of the underlying codes. In certain decoding schemes, such asmultistage or successive cancellation decoding, the lattice decoding processdirectly incorporates the decoders of the individual codes. This dependencyinherently restricts the choice of component codes, as their decoding effi-ciency and compatibility become critical to the overall system performance.Second, the computational efficiency of encoding and decoding proce-dures plays a pivotal role. Simpler and more structured codes often lead864.2. Construction D Lattices over Ring of Integersto significant reductions in implementation complexity, which is especiallydesirable for practical communication systems.Third, and perhaps most fundamentally, the characteristics of the com-munication channel significantly affect code selection. In well-studied sce-narios such as the AWGN channel, there exists a clear statistical modelat each decoding stage, which facilitates a principled approach to code de-sign. For example, the multilevel lattice framework in [23] relies on effectivesub-channels generated through coset partitioning, and component codes arespecifically designed to match the error characteristics of these sub-channels,allowing performance to approach theoretical limits under multistage decod-ing.In contrast, block-fading (BF) channels present a fundamentally differentchallenge. The non-ergodic nature of these channels results in zero Shan-non capacity, and the lack of a well-defined sub-channel structure at eachcoding level prevents straightforward application of classical code design cri-teria. Due to the absence of a predictable error distribution across levels, itbecomes difficult to match codes optimally to the fading behavior.Given these complications, this work adopts a deliberately channel-agnosticapproach. We select nested linear codes at fixed rates without aligning themto any particular fading profile or outage condition. This allows us to focuson the intrinsic behavior of algebraically constructed lattices, independentlyof channel-specific optimizations. The objective is to highlight the algebraicconstructionâs feasibility and structural properties rather than to maximizeperformance on a particular channel. Designing tailored coding strategiesthat account for the irregularities of block-fading channels remains an im-portant direction for future research.874.2. Construction D Lattices over Ring of IntegersFurthermore, to facilitate efficient encoding and decoding within this al-gebraic lattice framework, we develop in Subsection 4.2.2 a semi-structuredgenerator matrix derived from the algebraic properties of number fields. Al-though we initially introduce a simplifying assumption that the ideal Pj0is principal with norm p, which provides a clear algebraic link between thecode generator matrices and the lattice basis, our decoding algorithm doesnot rely on this connection. Instead, decoding is carried out directly on thelattice generator matrix without explicit reference to the underlying codestructure. This observation suggests that while the connection to compo-nent codes can simplify certain theoretical constructions, it is not essentialfor practical decoding. Future work could explore decoders that exploit thesealgebraic links to potentially improve efficiency or performance in more spe-cialized scenarios.Example: Construction D over K = Q(i)We illustrate Construction D with the number field K = Q(i), the fieldof Gaussian rationals. The ring of integers of K is OK = Z[i], which isan Euclidean domain and thus a principal ideal domain. Moreover, it ismonogenic with generator i.Let us choose the rational prime p = 5. In Z[i], we have the factorization:5 = (2 + i)(2â i),so the ideal â¨5â© splits as â¨5â© = P1 Â·P2, where P1 = â¨2+ iâ© and P2 = â¨2â iâ©are distinct prime ideals of norm N(P1) = N(P2) = 5.We takePj0 = â¨Î±j0â© = â¨2+iâ©. SincePj0 is principal andN(Pj0) = 5, theconditions required for Construction D are satisfied. Also, since OK/Pj0 â¼=884.2. Construction D Lattices over Ring of IntegersF5, the quotient Ptâ1j0 /Ptj0is isomorphic to F5 as a vector space.We now define a simple example of nested linear codes over F5 with blocklength N = 2. Let:â C1 â F25 be a [2, 1] linear code generated by the matrix M1 =[1 1].Its codewords are:C1 = {(0, 0), (1, 1), (2, 2), (3, 3), (4, 4)}.â C2 = F25 is the full ambient space, which is a [2, 2] linear code generatedby the identity matrix:M2 =ï£®ï£°1 10 1ï£¹ï£» .Clearly, C1 â C2, and both codes satisfy the nesting requirement forConstruction D.Using the ideal Pj0 = â¨2 + iâ©, we define the maps Ïi as:Ï1 : Z[i]â Z[i]/â¨2 + iâ©, Ï2 : Z[i]â â¨2 + iâ©/â¨(2 + i)2â©,and their component-wise extensions Φ1,Φ2, along with isomorphisms Ψ1,Ψ2as previously described.Then, the Construction D lattice Î â Z[i]2 is defined as:Î ={x â Z[i]2 â£â£Î¨t (Φt(x)) â Ct for t = 1, 2} .This lattice Î encodes the two layers of code constraints using reductionsmodulo Ptj0 , effectively lifting the codewords in C1, C2 to the arithmetic894.2. Construction D Lattices over Ring of Integersof Z[i], and yielding a structured discrete set in C2 through the canonicalembedding.4.2.2 Generator Matrix of Algebraic Construction DLatticesIn this subsection, we derive an explicit generator matrix for latticesconstructed via Algebraic Construction D. By lifting nested linear codes overFp through successive ideal quotients in the ring of integers OK , we obtaina structured basis that reflects both the algebraic and combinatorial layersof the construction. This generator matrix plays a key role in characterizingthe lattice geometry and facilitates applications such as encoding, decoding,and computing lattice invariants.Let p be a prime number, and suppose we are given a sequence of nestedlinear codes C1, C2, . . . , Ca over the finite field Fp, such that C1 â C2 â· · · â Ca. Each code Ci has parameters [N, ki], where i = 1, . . . , a. LetM denote a generator matrix for the largest code in this chain, namely Ca.Due to the nested structure of the codes, it is possible to organize M in aform that clearly captures the hierarchical relationships among them. Morespecifically, M can be arranged in a structured format that encodes theseinclusion properties.M =ï£®ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£°Ik1 0k1Ãâk2 0k1Ãâk3 · · · 0k1Ãâka B10âk2Ãk1 Iâk2 0âk2Ãâk3 · · · 0âk2Ãâka B20âk3Ãk1 0âk3Ãâk2 Iâk3 · · · 0âk3Ãâka B3.......... . .......0âkaÃk1 0âkaÃâk2 0âkaÃâk3 · · · Iâka Baï£¹ï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£»,904.2. Construction D Lattices over Ring of Integerswhere âki â ki â kiâ1, and each Bi is a matrix of size âki Ã (N â ka) fori = 1, . . . , a, with k0 = 0. The first k1 rows of this matrix generate the codeC1, the next k2 rows generate the code C2, and so on, up to the code Ca.Given a number field K of degree n, there exist n distinct embeddingsÏt : K â C that reflect the fieldâs algebraic structure. The ring of integersOK , being a free Z-module of rank n, admits a basis {w1, w2, . . . , wn} overZ. By applying each of these embeddings to the basis elements, one obtainsa lattice in Cn, where the structure of the lattice is determined by the actionof the embeddings on OK . The matrix whose rows are the images of thebasis elements under the embeddings serves as a generator matrix for thislattice, denoted Ï(OK), and is given explicitly by:T1 =ï£®ï£¯ï£¯ï£¯ï£°Ï1(w1) Ï2(w1) · · · Ïn(w1)....... . ....Ï1(wn) Ï2(wn) · · · Ïn(wn)ï£¹ï£ºï£ºï£ºï£» . (4.12)Let {µ1, . . . , µn} be a Z-basis for the ideal Pj0 , where each element µican be expressed as a linear combination of the integral basis elements wj ,namely,µi =nâj=1µi,jwj for i = 1, . . . , n.The image of this ideal under the canonical embedding Ï, denoted Ï(Pj0),forms a lattice in Cn. A generator matrix for this lattice, denoted by T2, is914.2. Construction D Lattices over Ring of Integersgiven by:T2 =ï£®ï£¯ï£¯ï£¯ï£°Ï1(µ1) Ï2(µ1) · · · Ïn(µ1)....... . ....Ï1(µn) Ï2(µn) · · · Ïn(µn)ï£¹ï£ºï£ºï£ºï£» =ï£®ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£°t1,1 t1,2 · · · t1,nt2,1 t2,2 · · · t2,n....... . ....tn,1 tn,2 · · · tn,nï£¹ï£ºï£ºï£ºï£ºï£ºï£ºï£» ,where ti,j =ânj=1 µi,jÏj(wj) for i, j = 1, . . . , n. The generator matrix of theConstruction D lattice Î, denoted by GÎ, is formally given byGÎ =ï£®ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£¯ï£°Ik1 âT1 0k1Ãâk2 âT1 · · · 0k1Ãâka âT1 B1 âT10âk2Ãk1 âT2 Iâk2 âT2 · · · 0âk2Ãâka âT2 B2 âT20âk3Ãk1 âT3 0âk3Ãâk2 âT3 · · · 0âk3Ãâka âT3 B3 âT3....... . .......0âkaÃk1 âTa 0âkaÃâk2 âTa · · · Iâka âTa Ba âTa0âka+1Ãk1 âTa+1 0âka+1Ãâk2 âTa+1 · · · 0âka+1Ãâka âTa+1 INâka âTa+1ï£¹ï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£ºï£»(4.13)In this formulation, the tensor product â is used to express the layeredtensor structure inherent to the construction. Each matrix Ti represents thegenerator matrix of the ideal Ï(Pij0), for i = 2, . . . , a + 1. These matricesare constructed analogously to T1 and T2, by determining a suitable Z-basis of the corresponding ideal Pij0 . The resulting matrix GÎ is squarewith dimensions nN ÃnN , ensuring that the lattice is full-rank and that allconstruction layers are coherently integrated.Proposition 4.1. Let K be a number field with signature (r1, r2) and Pj0 =αj0OK be a prime ideal with norm N(Pj0) = p. The Construction D latticeÎ based on number field K, defined in (4.11), is a sublattice of Ï(ONK) withdiscriminant â(Î) =(â|âK |2r2)2Np2aNâ2âai=1 ki.924.2. Construction D Lattices over Ring of IntegersProof. To simplify the presentation, we assume without loss of generalitythat a = 2. That is, we consider only two nested codes, C1 and C2, withparameters [N, k1] and [N, k2], respectively. The extension to arbitrary afollows directly from this case.Define the integer vectors:u1 = (u1,1, . . . ,u1,k1), u2 = (u2,1, . . . ,u2,k2âk1),andu3 = (u3,1, . . . ,u3,Nâk2),where each component ui,j â Zn.Let T1, T2, and T3 denote the generator matrices corresponding to thelattices constructed over the embedded ring Ï(OK) and the images of theideals Ï(Pj0) and Ï(P2j0), respectively. Then, we have(u1,u2,u3)ï£®ï£¯ï£¯ï£¯ï£°Ik1 âT1 0âT1 B1 âT10âT2 Ik2âk1 âT2 B2 âT20âT3 0âT3 INâk2 âT3ï£¹ï£ºï£ºï£ºï£»=[u1(Ik1 âT1),u2(Ik2âk1 âT2),u1(B1 âT1) + u2(B2 âT2) + u3(INâk2 âT3)]=[(u1,1T1, . . . ,u1,k1T1) ,(u2,1T2, . . . ,u2,(k2âk1)T2),u1(B1 âT1) + u2(B2 âT2) +(u3,1T3, . . . ,u3,(Nâk2)T3) ].Using the definition of T1,T2,T3, for i = 1, . . . , k1, we have u1,iT1 âÏ(OK), and for j = 1, . . . , âk2, u2,jT2 â Ï(Pj0) and for t = 1, . . . , N â k2,we have u3,tT3 â Ï(Pj0)2. Now consider Bl = (bli,j), for l = 1, 2, 3, which934.2. Construction D Lattices over Ring of Integersgives us(u1,u2,u3)GÎ =[(u1,1T1, . . . ,u1,k1T1), (u2,1T2, . . . ,u2,(k2âk1)T2),( k1âi=1b1i,1u1,iT1, . . . ,k1âi=1b1i,(Nâk2)u1,iT1)+( k2âk1âi=1b2i,1u2,iT2, . . . ,k2âk1âi=1b2i,(Nâk2)u2,iT2)+(u3,1T3, . . . ,u3,(Nâk2)T3)].We apply Ïâ1 on the resulting vector and denote the obtained vector byx = (x1,x2,x3) â Ok1K ÃPk2âk1j0 ÃONâk2K which has the following form:x =((x1,1, . . . , x1,k1), (x2,1, . . . , x2,k2âk1),( k1âi=1b1i,1x1,i, . . . ,k1âi=1b1i,(Nâk2)x1,i)+( k2âk1âi=1b2i,1x2,i, . . . ,k2âk1âi=1b2i,(Nâk2)x2,i)+(x3,1, . . . , x3,Nâk2)).Since the first k2 components of x are arbitrary elements ofOK andOK/Pj0 â¼=Fp, applying the homomorphism Φ1 to x, which computes modulo Pj0component-wise, yields x¯ = Φ1(x) â C1 in the following form:x¯ =((x¯1,1, . . . , x¯1,k1), (0 . . . , 0),944.2. Construction D Lattices over Ring of Integers( k1âi=1b1i,1x¯1,i, . . . ,k1âi=1b1i,(Nâk2)x¯1,i))= (x¯1,1, . . . , x¯1,k1)[Ik1 0(k2âk1) B1],where the last two summands in the third component of x diminish whenapplying Φ1 because their components belong to Pj0 and P2j0, which isa subset of Pj0 . It is evident that Ψ1 behaves as the identity map andsatisfies Ψ1 (Φ1(x)) â C1. By applying Φ2, we obtain the vector x¯ = (xâ x¯)(mod P2j0), which belongs to(Pj0/P2j0)N. Consequently, applying Ψ2 mapsthis vector to an element in FNp . Furthermore,x¯ =((x¯1,1, . . . , x¯1,k1), (x¯2,1, . . . , x¯2,k2âk1),( k1âi=1b1i,1x¯1,i, . . . ,k1âi=1b1i,(Nâk2)x¯1,i)+( k2âk1âi=1b2i,1x¯2,i, . . . ,k2âk1âi=1b2i,(Nâk2)x¯2,i))= (x¯1, x¯2)ï£®ï£° Ik1 0 B10 Ik2âk1 B2ï£¹ï£» ,for which we have Ψ2(Φ2(x)) = Ψ2(x¯) â C2, as needed. As we know,| det(T1)| = vol(Ï(OK)) = 2âr2â|âK |, therefore, the absolute value of954.2. Construction D Lattices over Ring of Integersthe determinant of GÎ can be computed as| det(GÎ)| = |det(Ik1 âT1)|aâi=2|det(Ikiâkiâ1 âTi)|Ã | det(INâka âTa+1)|= |det(T1)|k1aâi=2|det(Ti)|kiâkiâ1Ã | det(Ta+1)|Nâka=(â|âK |2r2)NN(Pj0)aNââai=1 ki=(â|âK |2r2)NpaNââai=1 ki ,where det(Ti) = vol(Ï(Piâ1j0)) =(â|âK |2r2)piâ1 for i = 2, . . . , a + 1. On theother hand, since Î â Ï(OK)N , let us consider L(Ï(OK)N ,Î) = Ï(OK)N â©V(Î). Similar to [54, Theorem 4], we can demonstrate that the elements ofthe following set C are linearly independent,C =ï£±ï£²ï£³aâ1âs=0ks+1âi=ks+1Î²(s+1)i Î¦â1s+1(Î¨â1s+1(bi))) | β(s+1)i â Zpaâsï£¼ï£½ï£¾ ,where k0 = 0, and {b1, . . . ,bka} is the basis of Ca. Hence,|C| =aâ1âs=0(p(aâs))(ks+1âks)= p(âai=1 ki),and the number of cosets is |L(Ï(OK)N ,Î)| = (pa)N/|C| = paNââai=1 ki .Additionally, we have |L(Ï(OK)N ,Î)| = |Ï(OK)N ||Î| =vol(Î)vol(Ï(OK)N ) . Hence,964.2. Construction D Lattices over Ring of Integersvol(Î) = paNââai=1 kivol(Ï(OK)N ) =(â|âK |2r2)NpaNââai=1 ki . This com-pletes the proof, as we have shown that the lattice generated by GÎ formsa sublattice of the Construction D lattice, based on the given family of un-derlying codes, with the same volume. Therefore, it must be equal to theentire lattice.Remark 4.2 (Information-Theoretic Perspective on Discriminant and Re-dundancy). From an information-theoretic perspective, the discriminant ex-pression in Proposition 4.1 provides a natural connection between codingredundancy and lattice geometry. In the absence of coding, the uncodedlattice Ï(OK)N represents the maximal entropy configuration, where all co-ordinates are freely selected over the ring of integers. The discriminant ofthis uncoded lattice is determined purely by algebraic number theory, givenbyâuncoded =(â|âK |2r2)2N,which corresponds to the square of the fundamental volume of Ï(OK)N .When nested linear codes C1 â · · · â Ca, with respective dimensionsk1, . . . , ka, are applied via Construction-D, the total symbol space expandsto aN symbols. However, onlyâai=1 ki symbols are information-bearing.The remaining aNââai=1 ki symbols introduce redundancy, imposing linearcongruence constraints on the lattice coordinates.From an information-theoretic standpoint, these redundant symbols re-duce the effective entropy of the system: each redundant symbol constrainsthe space by a factor of 1/p, thereby limiting the number of admissible latticepoints. Geometrically, these constraints enlarge the fundamental volume ofthe lattice by a factor of p per redundant symbol. Since the discriminant974.2. Construction D Lattices over Ring of Integersis proportional to the square of the fundamental volume, each redundantsymbol increases the discriminant by a factor of p2.Consequently, the discriminant of the coded lattice becomesâ(Î) = âuncoded · p2(aNââai=1 ki) =(â|âK |2r2)2Np2aNâ2âai=1 ki .This expression captures how coding reduces entropy while expanding thelattice volume and scaling the discriminant. It aligns precisely with thealgebraic derivation, offering a complementary viewpoint: the more redun-dancy introduced by coding, the larger the fundamental volume and thehigher the discriminant. This perspective highlights the dual role of codingin controlling both the information rate and the geometric density of latticepoints.Example 4.3. Let p = 5, and consider the cyclotomic field K = Q(ξ5),where ξ5 is a primitive 5th root of unity. Its totally real maximal subfield isK+ = Q(θ), with θ = ξ5 + Î¾â15 . The ring of integers of K+ is OK+ = Z[θ].The minimal polynomial of θ is f(x) = x2 + xâ 1, and hence the extensiondegree is [K+ : Q] = pâ12 = 2.The canonical embeddings Ïi : K+ â C for i = 1, 2 are given by:Ï1(Î¸) = θ =â5â 12,Ï2(Î¸) = θ¯ =ââ5â 12.Since {1, θ} forms a Z-basis of the ring of integers OK+ , the generator matrix984.2. Construction D Lattices over Ring of IntegersT1 of the embedded lattice is:T1 =ï£®ï£°1 1θ Î¸¯ï£¹ï£» . (4.14)The discriminant of K+ is given by âK+ = | det[Ïi(Î¸j)]|2 = 5. Since K+is totally real, its signature is (r1, r2) = (2, 0), which implies full diversityL = r1 = 2.Now, the ideal â¨5â© is completely ramified in OK+ , meaning that:5OK+ = P2j0 ,where Pj0 is a principal ideal generated by αj0 = 2 â θ. The norm of thisideal is N(Pj0) = N(αj0) = 5, and one can easily verify that â¨Î±ij0â© = â¨Î±j0â©i.The Z-bases for the ideals Pj0 and P2j0 are given by the sets {5, 3 + θ}and {5, 5θ}, respectively. Hence, the generator matrices for the embeddingsÏ(Pj0) and Ï(P2j0), denoted by T2 and T3, are:T2 =ï£®ï£¯ï£° 5 55 +â525ââ52ï£¹ï£ºï£» , (4.15)T3 =ï£®ï£¯ï£° 5 5â5 + 5â52â5â 5â52ï£¹ï£ºï£» , (4.16)with determinants |T2| = â5â5 and |T3| = â25â5, respectively.The ring homomorphismsΦi : ONK+ â(Piâ1j0Pij0)N, for i = 1, 2,994.2. Construction D Lattices over Ring of Integersare defined as:Φi(x) =[xâ(x mod Piâ1j0)]mod Pij0 .We also define Ψi :(Piâ1j0 /Pij0)N â FNp for i = 1, 2 as the component-wiseexpansion of the isomorphism Ïi : Piâ1j0/Pij0 â Fp. These maps allow us toassociate the quotient ideals with vectors in FNp , making it possible to verifywhether a given x â ONK satisfies the membership condition Ψi(Φi(x)) â Ci.Now, consider a pair of nested linear codes over F5, given by:C1 = â¨(100)â©, C2 = â¨(100), (011)â©.The Construction D lattice Î is defined as:Î ={λ â O3K+ | Ψi (Φi(λ)) â Ci, for i = 1, 2},with the corresponding generator matrix:GÎ =ï£®ï£¯ï£¯ï£¯ï£°1âT1 0âT1 0âT10âT2 1âT2 1âT20âT3 0âT3 1âT3ï£¹ï£ºï£ºï£ºï£»=ï£®ï£¯ï£¯ï£¯ï£°T1 0 00 T2 T20 0 T3ï£¹ï£ºï£ºï£ºï£» .Finally, using the block upper-triangular structure of GÎ, the volume of the1004.2. Construction D Lattices over Ring of Integerslattice is computed as:vol(Î) = |GÎ| = |T1||T2||T3| = |(ââ5)(â5â5)(â25â5)| = 54â5,which matches the theoretical prediction provided in Proposition 4.1.The generalized Construction D framework presented in this work ex-tends the classical Construction D, originally defined over the integer ringZ, to lattices constructed over the ring of integers OK of an arbitrary num-ber field K. In the classical setting, lattices are formed from a chain ofnested linear codes over a finite field Fp, and the resulting generator matrixtypically consists of scaled identity matrices and code components, pro-ducing a lattice in RN . While this approach is conceptually simple andefficient, it is limited in flexibility and algebraic richness. In contrast, theproposed construction operates over OK , incorporating the arithmetic ofprime ideal powers Pij0 and embedding the resulting algebraic elements intoEuclidean space via canonical embeddings. This leads to lattices in RnN ,where n = [K : Q], with enhanced structural richness and the potentialfor full diversity. The introduced semi-systematic generator matrix retainsthe hierarchical block structure of classical Construction D, but generalizesit using tensor products with ideal embedding matrices, thereby encodingboth the code structure and the arithmetic layering imposed by the num-ber field. This generalized framework enables more modular constructions,greater control over lattice volume via the field discriminant, and opens thedoor to applications in areas such as coding theory, signal processing, andcryptographic lattice design.With the structural and algebraic foundations in place, we now turn ourattention to the application of the proposed Construction D lattices in com-1014.3. Block Fading Channel Model and Performance Metricsmunication over block fading channels. In such channels, the transmittedsignal experiences piecewise-constant fading, making diversity and robust-ness critical factors in performance. The algebraic richness of lattices con-structed over OK , combined with their layered generator matrix structure,provides inherent advantages in combating the impairments introduced byfading blocks. In particular, the semi-systematic generator matrix enablesa natural extension of multistage decoding techniques to the number fieldsetting, allowing for efficient and structured recovery of lattice points underfading conditions.4.3 Block Fading Channel Model andPerformance MetricsIn this section, we consider the application of lattice codes over numberfields in the context of communication over real-valued block fading (BF)channels. Block fading models are a realistic wireless scenario in whichthe channel coefficients remain constant over fixed-length blocks and varyindependently between them. Unlike ergodic fading models, the block fadingchannel is non-ergodic, meaning that channel variations cannot be averagedout over time. As a result, performance is fundamentally limited by thequality of the instantaneous fading realization, and the error probabilitycannot vanish entirely, even with infinite code length.4.3.1 Channel ModelIn order to evaluate the performance of the proposed lattice construc-tions in realistic communication scenarios, we consider a flat block fading1024.3. Block Fading Channel Model and Performance Metricschannel model. This model is widely used in wireless communications, par-ticularly in systems where the signal bandwidth is narrow compared to thecoherence bandwidth of the channel. Under such conditions, all frequencycomponents of the transmitted signal experience the same fading effect,which remains constant over a finite number of symbol transmissions andchanges independently between blocks.Let xi â Rn denote the transmitted signal vector in the i-th fading block,where i = 1, . . . , N , n is the dimension of each transmitted vector (e.g.,corresponding to the number of transmit antennas or lattice dimension),and N is the number of independently fading blocks over the transmission.Each symbol undergoes flat fading across its n dimensions, modeled by adiagonal fading matrix Hf = diag(h) â RnÃn, where h = (h1, . . . , hn)contains the fading coefficients. The received signal yi â Rn is then givenby the relationyti = Hfxti + ni, (4.17)where ni â Rn is an additive white Gaussian noise (AWGN) vector withindependent entries distributed as N (0, Ï2N ). The same fading matrix Hfapplies to all vectors within the block, and a new independent realizationoccurs for each block.We consider Rayleigh fading channels as the underlying communicationmodel, which are appropriate in scenarios where the transmitted signal un-dergoes rich scattering with no dominant line-of-sight component. In suchenvironments, the received signal is a superposition of many independentlyscattered paths. By the central limit theorem, the resulting complex base-band channel coefficients hi can be modeled as zero-mean complex Gaussianrandom variables with variance Ï2h, i.e., hi â¼ CN (0, Ï2h).1034.3. Block Fading Channel Model and Performance MetricsUnder this model, the magnitude |hi| of the complex gain follows aRayleigh distribution, with probability density functionf|h|(x) =xÏ2hexp(â x22Ï2h), x â¥ 0, (4.18)where Ï2h denotes the variance of the fading process. Each complex gaincan be expressed in polar form as hi = aiejθi , where ai = |hi| is Rayleighdistributed and θi â¼ U [0, 2Ï) is uniformly distributed.When ideal phase tracking is available at the receiver, or when employingin-phase/quadrature (I/Q) interleaving techniques [40], the random phase θican be effectively neutralized. This allows the fading process to be modeledusing only the amplitude |hi|, resulting in a real-valued diagonal fadingmatrix HF = diag(|h1|, . . . , |hn|), where the |hi| are independent Rayleigh-distributed random variables.Therefore, to describe the received signal across the entire transmissionframe, we considerN transmitted symbol vectors, forming x = (x1, . . . ,xN ) âRnN . Stacking the corresponding received vectors and noise terms yields theoverall input-output relation:yt = (IN âHF )xt + nt, (4.19)where n = (ν1, . . . , νnN ) â RnN is the noise vector composed of indepen-dent Gaussian samples νi â¼ N (0, Ï2N ), for i = 1, . . . , nN . The signalvectors xi are drawn from a lattice Î â RnN constructed via ConstructionD over a number field K of degree n. Each component xi corresponds to asymbol constructed from nested linear codes C1 â · · · â Ca, with Ci beingan [N, ki]-linear code over Fp. This structure supports communication over1044.3. Block Fading Channel Model and Performance Metricsblock fading channels with N symbols per block and n independently fadeddimensions.To evaluate the performance of such systems, a key parameter is theinstantaneous signal-to-noise ratio (SNR), which quantifies the power of thetransmitted lattice point relative to the noise level. For an infinite latticeconstellation Î, the SNR is given byγ =vol(Î)2nNÏ2N, (4.20)where vol(Î) denotes the volume of the fundamental region of the lattice,and Ï2N is the noise variance per real dimension. This definition reflects thedecoding threshold for noise-limited performance in the absence of powerconstraints.4.3.2 Error Performance over BF ChannelsThe error behavior of lattice codes in BF channels is strongly influencedby the nature of the fading process. In particular, the non-ergodic nature ofthe block-fading channelâwhere the fading remains constant over each blockand varies independently across blocksâintroduces a fundamental perfor-mance limitation: the possibility of outage events. These occur when the in-stantaneous channel conditions cannot support the target transmission rate,regardless of coding strategy. One might ask whether encoding across thespatial dimension (i.e., across the n components of each transmitted vector)with nââ could lead to ergodic behavior. However, in the model consid-ered here, the fading coefficients h1, . . . , hn remain fixed during each blocktransmission and are only updated across blocks. Thus, even as nââ, thechannel remains non-ergodic because coding is restricted within blocks of1054.3. Block Fading Channel Model and Performance Metricsfixed fading realizations. This differs from the classical ergodic fading sce-nario, where the code spans many independent fading realizations, allowingthe transmitter and receiver to exploit channel averaging to mitigate deepfades. In contrast, our framework considers finite block-length transmis-sion under block-wise constant fading, making outage events a fundamentalconstraint on performance. . In such scenarios, the instantaneous channelconditions may be so poor that decoding fails, regardless of the SNR orcode length. This motivates the need for lattice constructions that not onlyprovide good distance properties but also exhibit robustness across a rangeof fading realizations.Consider a lattice Î â RnN generated by a full-rank matrix GÎ âRnNÃnN , so thatÎ = {uGÎ | u â ZnN}.The transmitted codeword x â Î is affected by BF channel coefficients,which scale each sub-block of the transmission. The received lattice gener-ator under a particular fading realization becomesGÎF = (IN âHF )GÎ,where HF = diag(|h1|, . . . , |hn|) is the diagonal matrix of fading magni-tudes, and IN is the identity matrix of size N . This formulation models nindependent fading blocks, each acting on a sequence of N components.Assuming perfect channel state information (CSI) at the receiver, de-coding can be performed with full knowledge of the fading coefficients. Dueto the geometric symmetry of lattice constellations, analysis typically as-sumes transmission of the all-zero vector. The decoding region, therefore,1064.3. Block Fading Channel Model and Performance Metricscorresponds to the Voronoi cell of the origin in the faded lattice:VÎ(x,h) ={y â RnN | â¥y â hxâ¥ â¤ â¥y â hwâ¥ âw â Î} .Due to the geometric uniformity of lattices, we have V(x,h) = V(w,h) =VÎ(h) for all x,w â Î, allowing us to assume the transmission of the all-zero lattice vector. The probability of decoding error using ML decoding,averaged over all channel realizations, is given byPe(Î, Ï2N ) = Eh[Pe(Î, Ï2N | h)]= 1â Eh[â«VÎ(h)(12ÏÏ2N)nN2exp(ââ¥yâ¥22Ï2N)dy],where Ï2N denotes the noise variance per dimension. Since the errorprobability is determined by the interplay of noise and lattice structure, itis often expressed in terms of the SNR, γ, and we will write Pe(γ) whereappropriate.A key performance indicator in this context is the diversity order, whichquantifies the rate at which the error probability decays with increasingSNR. Specifically, it is defined asd = â limÎ³ââlogPe(Î³)log γ. (4.21)A higher value of d corresponds to stronger resistance against deep fades.For a lattice to achieve full diversity in an n-block BF channel, this ordermust equal the number of independently faded components.Definition 4.4. A lattice Î â RnN is said to achieve full diversity if itsdiversity order satisfies d = n.1074.3. Block Fading Channel Model and Performance Metrics4.3.3 Error Probability and Design Criteria for Latticesover Block-Fading ChannelsHaving introduced the system model for lattice-coded communicationover block-fading (BF) channels in the previous section, we now focus onthe derivation of error probability bounds and the corresponding designprinciples for reliable lattice constructions [29].Let Î â RnN be an nN -dimensional lattice used for transmission. Thedecoder selects the lattice point w â Î that minimizes the Euclidean de-coding metric conditioned on the channel realization h. A decoding erroroccurs if a point w Ì¸= x is chosen instead of the transmitted point x â Î.The point error probability Pe(Î, Ï2N | x) is the probability that thisdecoding error occurs. Applying the union bound, it can be upper-boundedas:Pe(Î, Ï2N | x) â¤âwâÎ{x}P (xâ w), (4.22)where P (xâ w) denotes the pairwise error probability (PEP)âthe proba-bility that the decoder selects w over x.At high signal-to-noise ratio (SNR), the PEP can be bounded using theChernoff technique [29], yielding:P (xâ w) â¤ 12âiâI8Ï2N(xi â wi)2 =(8Ï2N)â2d(â)p (x,w)2, (4.23)where:â I = {i â {1, . . . , nN} | xi Ì¸= wi} is the set of differing coordinates,â â = |I| is the number of such coordinates,â d(â)p (x,w)2 =âiâI(xi â wi)2 is the â-product distance.1084.3. Block Fading Channel Model and Performance MetricsDue to the geometric uniformity of lattices, the point error probabilityis independent of the transmitted vector:Pe(Î, Ï2N | x) = Pe(Î, Ï2N ), âx â Î. (4.24)Accordingly, one may assume x = 0 for analytical simplicity. Nevertheless,the general bound in (4.23) highlights three key structural properties of thelattice that govern its error performance under block fading.1. Diversity Order LL = minx Ì¸=wâÎ|{i | xi Ì¸= wi}| . (4.25)The diversity order measures the minimum number of independently fad-ing components over which any two lattice vectors differ. A lattice achievesfull diversity if L = n, where n is the number of fading blocks. Full diver-sity ensures resilience against fading by distributing information across allindependent channel realizations.2. Minimum L-Product Distanced(L)p,min = minx Ì¸=wâÎ|{i|xi Ì¸=wi}|=Lâi|xi Ì¸=wi|xi â wi|. (4.26)This metric captures the geometric separation between the closest latticepoint pairs that differ in exactly L dimensions. A higher value of d(L)p,minimproves the latticeâs robustness by reducing the pairwise error probabilityat high SNR.1094.3. Block Fading Channel Model and Performance Metrics3. Product Kissing Number ÏpThe product kissing number counts the number of lattice vectors at theminimum L-product distance from a given point. A smaller Ïp reduces thenumber of dominant error events contributing to the union bound, therebyimproving decoding performance.Design ImplicationsTogether, the diversity order L, the minimum L-product distance d(L)p,min,and the product kissing number Ïp provide a comprehensive framework forevaluating and designing lattices for BF channels.â Maximize the diversity order L,â Maximize the minimum product distance d(L)p,min,â Minimize the product kissing number Ïp.To compare the robustness of different lattice constructions, a normalizedversion of the product distance is often used:Ndp,min(Î) =dLp,min(Î)vol(Î).In the case of algebraic lattices derived from number fields, this expressionis closely tied to the field discriminant. For a totally real number field K ofdegree n, and its canonical embedding Ï(OK), we have [59]vol(Ï(OK)) =â|âK |, and Ndp,min(Ï(OK)) =1â|âK | .This result highlights the advantage of choosing number fields with smalldiscriminants in lattice design. Such fields give rise to dense, well-separated1104.3. Block Fading Channel Model and Performance Metricslattice structures that are inherently more resilient to the degradations im-posed by BF channels.4.3.4 Poltyrev Threshold and Outage Limit inLattice-Coded Block-Fading ChannelsUnderstanding the fundamental performance bounds of lattice-codedsystems over BF channels is essential for reliable transmission design. Twokey metrics that characterize these bounds are the Poltyrev threshold andthe Poltyrev outage limit (POL). These constructs provide valuable insightinto the error performance and achievable diversity of lattice codes undernon-ergodic fading scenarios.Poltyrev ThresholdThe Poltyrev threshold quantifies the maximum noise variance a lattice-coded system can tolerate for a given fading realization without incurringan unbounded probability of decoding error. In BF channels, this thresholdserves as a benchmark for instantaneous channel conditions. For an instan-taneous fading vector h = (h1, h2, . . . , hn), the Poltyrev threshold is givenby:Ï2max(h) =|det(GÎ)| 2nNâni=1 h2ni(2Ïe), (4.27)where GÎ is the generator matrix of the lattice Î, N denotes the lattice di-mension per fading block, and n is the number of such blocks. For successfuldecoding, the channel noise variance Ï2N must satisfy:Ï2N < Ï2max(h). (4.28)1114.3. Block Fading Channel Model and Performance MetricsThis condition ensures that the noise does not overwhelm the decoding capa-bilities of the lattice. The threshold inherently depends on both the latticestructure (through the determinant of the generator matrix) and the instan-taneous quality of the fading coefficients. It thus establishes a key designconstraint for achieving near-capacity performance in fading environments.Poltyrev Outage Limit (POL)The Poltyrev outage limit extends the concept of the Poltyrev thresholdto a probabilistic framework by quantifying the likelihood that a randomfading realization renders reliable communication infeasible. Introduced in[45], the POL provides a lower bound on the frame error probability dueto deep fades and is an essential metric for characterizing diversity. It hasbeen analytically shown that the POL achieves full diversity order L forchannels with L independently faded blocks, highlighting its optimality indiversity-limited regimes. For a given SNR γ and fading vector h, the outageprobability is expressed as:Pout(γ) = Prï£«ï£­Ï2N > |det(GÎ)| 2nN âni=1 h 2ni(2Ïe)ï£¶ï£¸= Pr(nâi=1h2i <(2Ïe)nÎ³n).(4.29)This expression captures the probability that the product of the squaredfading gains falls below a critical SNR-dependent threshold. An outage isdeclared when this condition is met, indicating that the channel cannotsupport reliable decoding under the given noise conditions. In practice,the frame error rate (FER) obtained through simulation or experimentation1124.4. Decoding of Full-Diversity Construction D latticescan be compared against Pout(γ) to estimate the SNR gap and verify theachieved diversity order. The POL thus offers a rigorous benchmark foranalyzing and optimizing the performance of lattice codes in block-fadingchannels.4.4 Decoding of Full-Diversity Construction DlatticesThis section presents a decoding strategy tailored for full-diversity lat-tices constructed via Construction D and designed to operate effectivelyover Rayleigh block fading channels. The primary decoding challenge insuch environments stems from the dual impact of additive Gaussian noiseand multiplicative fading. The goal is to exploit the diversity inherent inthe lattice structure while ensuring practical decoding complexity.Construction D lattices possess a natural multi-level architecture, mak-ing them well-suited for a layered decoding approach. We propose a schemethat combines ML decoding in the initial layers with successive cancellation(SC) in the remaining ones. This hybrid method aligns with the hierarchicalstructure of the lattice and the block fading model to achieve both decodingreliability and computational efficiency.Let us consider a Rayleigh block fading channel with n independentlyfading subchannels and a coherence interval of N . The received signal vectory, defined in Equation (4.19), contains the faded and noisy version of thetransmitted lattice codeword x. The decoding process aims to reconstructx by estimating a structured vector uË = (uË1, . . . , uËN ), where each subvectoruËj = (uËj,1, . . . , uËj,n) corresponds to the jth coherence block.The decoding procedure is divided into two stages, reflecting the layered1134.4. Decoding of Full-Diversity Construction D latticesnature of Construction D. In the first stage, the initial ka subvectors uËj ,for j = 1, . . . , ka, are recovered using ML decoding. This is implementedthrough lattice quantization:uËj = QÎâ²j (ytj), (4.30)where yj = y((j â 1)n+ 1 : jn) extracts the segment of the received vectorcorresponding to the jth block. The decoding lattice Îâ²j is generated by thematrix Tâ²j = HFGtj,j , with Gj,j denoting the n Ã n submatrix of the fullgenerator matrix GÎ, extracted using indices (j â 1)n+ 1 to jn.Once the most critical ka layers are decoded, we proceed with successivecancellation to recover the remaining subvectors uËka+1, . . . , uËN . The contri-butions of the already decoded components are subtracted from the receivedsignal to isolate the residual information:yËâ² = yj âkaât=1uËtGt,tHF . (4.31)The adjusted signal yËâ² is then quantized using the same lattice decodingprocedure to retrieve the remaining components. This full decoding processis summarized in Algorithm 1.1144.4. Decoding of Full-Diversity Construction D latticesAlgorithm 1 Full-Diversity Construction-D Lattices Decoding1: procedure DEC(y,GÎ,HF = diag(|h1|, . . . , |hn|))2: uËâ 01ÃnN3: xËâ 01ÃnN4: for j = 1 : N do5: yj â y((j â 1) · n+ 1 : j · n)6: Gj,j â(GÎ(i, k))i,kâ{(jâ1)·n+1,...,j·n}7: if j â¤ ka then8: uËj â argminujâZn â¥yj â ujGj,jHF â¥29: uË((j â 1) · n+ 1 : j · n)â uËj10: xË((j â 1) · n+ 1 : j · n)â uËjGj,j11: else12: yj â yj ââkat=1 uËtGt,tHF13: uËj â argminujâZn â¥yj â ujGj,jHF â¥214: uË((j â 1) · n+ 1 : j · n)â uËj15: xË((j â 1) · n+ 1 : j · n)â uËjGj,j16: end if17: end for18: return uË, xË19: end procedureExample 4.5 (Step-by-Step Decoding of a Construction-D Lattice). Con-sider the totally real quadratic field K = Q(â10), with ring of integersOK = Z[â10]. The canonical embedding yields the generator matrices:T1 =ï£®ï£° 1 1â10 ââ10ï£¹ï£» , T2 =ï£®ï£° 2 2â10 ââ10ï£¹ï£» ,T3 =ï£®ï£° 2 22â10 â2â10ï£¹ï£» .1154.4. Decoding of Full-Diversity Construction D latticesLet n = 2, N = 3, a = 2, and consider the binary code generator matrixM =ï£®ï£°1 0 10 1 1ï£¹ï£» .with Construction-D lattice generator matrixGÎ =ï£®ï£¯ï£¯ï£¯ï£°T1 0 T10 T2 T20 0 T3ï£¹ï£ºï£ºï£ºï£» .Assume a fading vector h = (1.3, 0.) and fading matrix HF = diag(1.3, 0.8).The transmitted information vector isu = (1, 0, 1, 1, 0, 1),which maps to the lattice pointx =(1, 1, 2 +â10, 2ââ10, 3 + 3â10, 3â 3â10).Let the additive noise vector ben = (0.01,â0.02, 0.05, 0.03,â0.04, 0.02) ,giving us the received signal yt = (I3 âHF )xt + nt, asy = (1.31, 0.78, 6.76099,â0.8998, 16.19297,â5.1695).Decoding proceeds using multistage successive cancellation as follows:1164.4. Decoding of Full-Diversity Construction D latticesLevel 1 (j = 1): Decode from y1 = y(1 : 2):uË1 = arg minuâZ2â¥y1 â uT1HF â¥2 â uË1 = [1, 0].Level 2 (j = 2): Decode from y2 = y(3 : 4):uË2 = arg minuâZ2â¥y2 â uT2HF â¥2 â uË2 = [1, 1].Level 3 (j = 3): First, subtract the prior contribution:yâ²3 = y(5 : 6)â uË1T1HF â uË2T2HF .Then decode as before:uË3 = arg minuâZ2â¥yâ²3 â uT3HF â¥2 â uË3 = [0, 1].The final decoded vector and reconstructed lattice point areuË = (1, 0, 1, 1, 0, 1), xË = uË Â·GÎ,which matches the original transmitted codeword. This example illustratesfull-diversity decoding under fading and additive noise using the proposedConstruction-D framework.4.4.1 Diversity Analysis of the Decoding AlgorithmThis subsection is dedicated to evaluating the diversity performance ofthe proposed decoder when applied to algebraic lattices over Rayleigh blockfading channels. The objective is to demonstrate that the decoding process1174.4. Decoding of Full-Diversity Construction D latticesachieves full diversity, meaning that the decoding error probability decaysproportionally to Î³ân, where γ represents the signal-to-noise ratio and n isthe number of fading blocks.Suppose that communication is carried out using either the lattice Î orone of the scaled embeddings Ï(Pij0), for i = 1, . . . , a. When used over ablock fading channel with n independently faded blocks, the SNRs associatedwith each structure are given byÎ³Î =vol(Î)2/(nN)Ï2N=(âN/2K paNââai=1 ki)2/(nN) · 1Ï2N, (4.32)Î³Pij0=vol(Ï(Pij0))2/nÏ2N=(piââK)2/n · 1Ï2N. (4.33)Assuming both settings achieve full diversity, the corresponding error prob-abilities are governed by the terms 1/Î³nÎ and 1/γnPij0. These expressions arerelated throughÎ³Î =(paâ1Nâai=1 kiâi)2/n · γPij0 , (4.34)which implies the asymptotic equivalenceO(1Î³Î)= O(1γPij0). (4.35)At high SNR values, when Ï2N â 0, both SNR expressions become effec-tively equivalent. For convenience, they are collectively denoted by γ in thefollowing analysis.Theorem 4.6. Let C1 â C2 â · · · â Ca â FNp be a sequence of nested linearcodes of parameters [N, ki], forming a Construction D lattice Î with diversityorder n. Then, when decoded using the algorithm described in Algorithm 1,1184.4. Decoding of Full-Diversity Construction D latticesthe lattice Î achieves full diversity over a block fading channel with n fadingblocks.Proof. Since the transmitted lattice point is generated from an integer vectoru, recovering x is equivalent to estimating u. The optimal decoding processinvolves solving the minimization problemuË = arg minuâZnNâ¥â¥yt â (uGÎ(IN âHF ))tâ¥â¥2 , (4.36)which corresponds to the closest vector problem (CVP) in dimension nN ,known to be NP-hard in the general case. To make decoding feasible inpractice, the proposed scheme reduces this high-dimensional problem to Nlower-dimensional problems in dimension n.According to the algorithm, the first ka segments of uË are recovered vian-dimensional maximum-likelihood decoding, as follows:(uË1, . . . , uËka) =kaâj=1arg minujâZnâ¥â¥ytj âHFGtj,jutjâ¥â¥2 , (4.37)where yj is the segment of y corresponding to the j-th block, andâdenotesconcatenation of the vectors. After these layers are decoded, the signalis updated and the remaining components are recovered using successivecancellation. For j = ka + 1, . . . , N , we defineyâ²j = yj âkaât=1uËtGt,tHF , (4.38)and perform(uËka+1, . . . , uËN ) =Nâj=ka+1arg minujâZnâ¥â¥â¥yâ²tj âHFGtj,jutjâ¥â¥â¥2 . (4.39)1194.4. Decoding of Full-Diversity Construction D latticesTo analyze the approximation gap, define wj â RnN as a zero vector exceptin the j-th segment, wherewj((j â 1)n+ 1 : jn) =ï£±ï£´ï£²ï£´ï£³ytj âHFGtj,jutj , for 1 â¤ j â¤ ka,yâ²tj âHFGtj,jutj , for j > ka.(4.40)The total decoding metric can then be written asâ¥â¥yt â (uGÎ(IN âHF ))tâ¥â¥2 =â¥â¥â¥â¥â¥â¥Nâj=1wjâ¥â¥â¥â¥â¥â¥2. (4.41)Applying the triangle inequality yieldsâ¥â¥â¥â¥â¥â¥Nâj=1wjâ¥â¥â¥â¥â¥â¥2â¤Nâj=1â¥wjâ¥2=kaâj=1â¥â¥ytj âHFGtj,jutjâ¥â¥2 + Nâj=ka+1â¥â¥â¥yâ²tj âHFGtj,jutjâ¥â¥â¥2 . (4.42)Note that each vector wj â RnN is constructed to have nonzero entriesonly in its own jth segment of length n, and is zero elsewhere. Therefore,the vectors wj have disjoint support, which implies that their pairwise innerproducts vanish:â¨wi,wjâ© = 0 for i Ì¸= j.As a result, the inequality in (4.42) actually holds with equality:â¥â¥â¥â¥â¥â¥Nâj=1wjâ¥â¥â¥â¥â¥â¥2=Nâj=1â¥wjâ¥2 .This justifies the validity of decomposing the global decoding error into a1204.4. Decoding of Full-Diversity Construction D latticessum of independent per-block errors, a structure enabled by the block fadingmodel and the segmented decoding strategy used in the algorithm.The decoder minimizes this upper bound rather than the exact ML met-ric, resulting in a suboptimal yet tractable solution.Since each n-dimensional decoding step achieves diversity order n, the errorprobability per segment is bounded by Î³ân. A point error in the finalestimate occurs if at least one segment is decoded incorrectly. Applyingthe union bound gives an overall error probability bounded by N · Î³ân,confirming that full diversity of order n is achieved.4.4.2 Complexity Analysis of the Decoding AlgorithmThis subsection examines the computational complexity of the decod-ing procedure introduced in Algorithm 1, which is composed of three mainoperations: an initial maximum-likelihood decoding phase, an intermediatestage involving successive cancellation, and a concluding ML decoding phasefor the remaining segments.The first stage applies ML decoding independently to the first ka latticesegments. The complexity of this phase can be expressed as O(ka · f(n)),where f(n) denotes the complexity of ML decoding in dimension n. Since ncorresponds to the number of fading blocks and is typically small, the decod-ing complexity in this dimension can be assumed polynomial and thereforecontributes a relatively minor computational burden compared to the scal-ing in ka. Hence, the complexity of this phase grows linearly with respectto ka.Following this, the decoder performs a cancellation step, in which theinfluence of the previously decoded segments is subtracted from the received1214.4. Decoding of Full-Diversity Construction D latticessignal before decoding the remaining components. Each subtraction involvesa matrix-vector multiplication with a matrix Gt,t â RnÃn. The complexityper operation is O(n2), and since this process is repeated ka times, the totalcomplexity of the cancellation phase is given by O(ka · n2).In the final stage, the decoder applies ML decoding to the remainingN â ka components. The total complexity of this step is O((N â ka) · f(n)),using the same per-segment complexity as in the first phase.Combining the contributions of all three stages, the overall decodingcomplexity is given by:C = O(ka · f(n)) +O(ka · n2) +O((N â ka) · f(n)). (4.43)Grouping the terms that depend on f(n), this simplifies to:C = O(f(n) Â·N) +O(ka · n2). (4.44)Given that n is a small constant relative to N , the second term becomesasymptotically negligible, particularly when N is large. Therefore, the totaldecoding complexity can be approximated as:C â O(N · f(n)). (4.45)This result shows that the overall computational complexity grows lin-early with the block length N , making the decoding algorithm scalable forhigh-dimensional lattice codes. The structure of the decoderâdividing thetask into parallelizable, low-dimensional ML problems and leveraging ef-ficient cancellationâcontributes significantly to this favorable complexityprofile.1224.4. Decoding of Full-Diversity Construction D latticesBy designing the decoder to operate efficiently in dimension n whilemaintaining full diversity, the scheme provides a practical trade-off betweenperformance and complexity. This ensures that the approach remains viablein real-world systems where both decoding reliability and computationalefficiency are critical.4.4.3 Parity-Check Enforcement and Its Impact onDiversityThe decoding architecture proposed in this work is tailored to leveragethe algebraic structure inherent in Construction-D lattices while maintainingcomputational efficiency through layered, multistage processing. However,a key limitation lies in the fact that the decoder does not explicitly enforcethat each decoded component belongs to its respective linear code Ci â FNp ,for i = 1, . . . , a.Although the decoding process utilizes the upper-triangular structure ofthe generator matrix to reconstruct lattice points, it does not verify that thedecoded codeword cËi = Î¨i(Î¦i(xË)) satisfies the corresponding parity-checkconditionHicËti â¡ 0. Given the semi-systematic form of the generator matrixintroduced in (4.13), the associated parity-check matricesHi for i = 1, . . . , atake the following structured forms:Hi =ï£®ï£¯ï£° INâki0(kaâki)ÃkiâBt1 âBt2 · · · âBtiâ1 âBtiï£¹ï£ºï£», (4.46)Ha =[INâkaâBt1 âBt2 · · · âBtaâ1 âBta]. (4.47)Incorporating these parity constraints into the decoding process would re-1234.4. Decoding of Full-Diversity Construction D latticesquire generating a list of the κ nearest candidate vectors uËj,1, . . . , uËj,Îº ateach decoding level j, instead of selecting only the nearest neighbor. De-coding would proceed from the deepest level j = a, verifying whether cËasatisfies HacËta â¡ 0. If satisfied, decoding continues upward. If the checkfails at any level, the decoder selects the next closest candidate and repeatsthe decoding process accordingly. This backtracking continues iterativelyuntil a consistent solution is found across all levels.While this enhanced method improves reliability, it is intentionally omit-ted from our current decoder to maintain low complexity. Enforcing paritychecks at each stage introduces additional operationsâsuch as syndromecomputations and re-decoding stepsâthat increase latency and underminethe modular and parallel nature of the layered decoder. Additionally, incor-porating such constraints complicates theoretical analysis of diversity per-formance. For large block lengths or high-rate codes, these operations maybe impractical in real-time communication scenarios.Nevertheless, this simplification has critical implications for the achiev-able diversity order. Suppose the final level code Ca has minimum Ham-ming distance dmin. Under canonical embedding, each symbol maps to ann-dimensional lattice block. Thus, any valid codeword of weight dmin yieldsa lattice point with at least dmin nonzero embedded componentsâeach ex-periencing independent fadingâleading to a maximum diversity of n · dmin.However, in the absence of parity enforcement, the decoder may ac-cept invalid vectors corresponding to low-weight (even weight-one) configu-rations, effectively reducing the minimum distance to dmin = 1, and limitingthe achievable diversity to n. This degradation is an inherent trade-off inour current decoding strategy, which prioritizes efficiency over strict codeconstraint enforcement.1244.5. Simulation ResultsTo address this, we are developing an extended decoding scheme thatintegrates parity-check validation within the multistage framework. By ver-ifying codeword membership through syndrome checks and applying correc-tions when necessary, the proposed enhancement aims to restore full diver-sity. Although this increases computational complexity, especially for high-rate or long codes, prior work [14, 29] has shown that such methodsâe.g.,belief-propagation and iterative decodingâcan significantly improve errorperformance at moderate SNRs. Our goal is to balance performance andcomplexity by exploiting the semi-systematic generator structure for par-tially joint decoding while preserving the scalability of the layered architec-ture.4.5 Simulation ResultsThis section provides a numerical evaluation of the proposed decodingmethod applied to full-diversity algebraic lattices over Rayleigh block fadingchannels. The performance is measured in terms of frame error rate (FER)across a range of signal-to-noise ratio (SNR) values. Here, SNR is definedasγ =vol(Î)2/(nN)Ï2N,where vol(Î) denotes the volume of the transmitted lattice and Ï2N is thenoise variance.In this work, the family of nested linear codes {Ci}ai=1 used in the latticeconstruction are generated randomly over the finite field Fp, with prescribedparameters [N, ki] for each level. No specific algebraic structure or optimiza-tion criteria are imposed on these codes beyond the required dimensionality1254.5. Simulation Resultsand nesting constraints. This design choice aligns with the philosophy ofthe proposed decoding framework, which interacts only with the generatormatrix of the overall Construction-D lattice and remains agnostic to theinternal structure of the component codes.Importantly, the decoding algorithm operates without requiring explicitknowledge of the individual codebooks or their generator matrices. As aconsequence, the decoderâs performance exhibits robustness with respect tothe particular realization of the nested codes, as long as the rate hierarchyand nesting are preserved. Simulation results confirm that replacing oneset of nested codes with anotherâdrawn randomly but with the same pa-rametersâdoes not significantly alter the frame error rate or the diversityperformance. This separation between code design and decoder operationenhances flexibility, allowing scalable implementations without reliance oncarefully optimized or structured codes.To assess how closely the decoder approaches theoretical performancelimits, we compare the FER results against the Poltyrev Outage Limit(POL), which represents a fundamental reliability bound for infinite lat-tice constellations under fading conditions [45]. The POL is sensitive toboth the fading distribution and the geometry of the lattice, particularly itsdeterminant, which is affected by the discriminant âK of the number fieldand the code rate parameters.Additionally, we include performance curves for full-diversity Construc-tion A lattices reported in [29] as a point of reference. This comparisonhighlights the relative efficiency of the proposed Construction D based ap-proach under identical channel conditions.Figure 4.1 presents the outage limits for Construction D lattices con-figured with various diversity orders and nested code structures. In this1264.5. Simulation Resultsfigure, the Poltyrev outage limits are plotted for Construction D latticeswith various diversity orders n and two discriminant values, âK = 1 andâK = 100. The curves demonstrate that increasing the diversity order leadsto a noticeable reduction in outage probability, confirming the theoreticalbenefit of diversity in fading environments. Additionally, lattices associatedwith smaller discriminants exhibit better outage performance, highlightingthe influence of lattice geometry on reliability.The POL serves as a theoretical bound on the probability of decodingfailure for infinite lattice constellations over fading channels. It dependson the channelâs fading statistics and the normalized volume of the lat-tice, which is influenced by âK and the underlying code rates. This plotprovides a benchmark for evaluating the effectiveness of practical lattice de-coders, particularly in the upcoming simulation results where performanceis compared against these limits.Figure 4.2 illustrates the decoding performance of several double-diversity(i.e., n = 2) Construction D lattices constructed over distinct real quadraticfields, compared against their corresponding POL. Each configuration usesnested linear codes with parameters [N, k2, k1] = [100, 50, 40], and decodingis performed using the layered algorithm described in Section 4.4.The lattices are built over three number fields: Q(â2), Q(â7), andQ(â10), representing a mix of low and moderate discriminant cases. ForQ(â2) and Q(â10), where m â¡ 2 (mod 4), the ring of integers is OK =Z[âm] with discriminants dK = 4m = 8 and dK = 40, respectively. Theprime ideals used in the lattice construction are P1 = 2OK +âmOKand P2 = 2OK + 2âmOK , yielding the respective Z-bases {2,âm} and1274.5. Simulation Results15 20 25 30 3510-610-510-410-310-210-1100Frame Error rate (FER)n=2,dK=1n=3,dK=1n=4,dK=1n=5,dK=1n=6,dK=1n=7,dK=1n=8,dK=1n=2,dK=100n=3,dK=100n=4,dK=100n=5,dK=100n=6,dK=100n=7,dK=100n=8,dK=100Figure 4.1: Poltyrev outage limit for full-diversity Construction-D latticeswith [N, k2, k1] = [100, 50, 40] and different diversity orders.{2, 2âm}. The resulting generator matrices used in decoding are:T1 =ï£®ï£° 1 1âm ââmï£¹ï£» , T2 =ï£®ï£° 2 2âm ââmï£¹ï£» .In contrast, for Q(â7), with m â¡ 3 (mod 4), the discriminant becomesdK = 28, and OK = Z[â7]. The prime ideals chosen are P1 = 2OK +(1 +â7)OK and P2 = 2OK + 2â7OK , with integral bases {2, 1 +â7} and1284.5. Simulation Results{2, 2â7}, respectively. This results in the generator matrices:T1 =ï£®ï£° 1 1â7 ââ7ï£¹ï£» , T2 =ï£®ï£° 2 21 +â7 1ââ7ï£¹ï£» , T3 =ï£®ï£° 2 22â7 â2â7ï£¹ï£» .The figure compares the simulated FER of these lattices against thetheoretical POL curves. At a target FER of 10â4, the Construction D lat-tice over Q(â10) exhibits a performance gap of approximately 7 dB fromits associated outage limit. This result improves upon the double-diversityConstruction A lattice over Q(â10) studied in [29], which reports a largergap of 8.45 dB under comparable parameters. This 1.45 dB improvementhighlights the benefits of the multi-level Construction D approach in exploit-ing both algebraic and structural diversity.Among the tested lattices, the one constructed over Q(â7) demonstratesthe most favorable performance, achieving only a 4.8 dB gap to the POL atthe same error rate threshold. This corresponds to a 2.2 dB advantage overthe Q(â10) lattice and underscores the effect of using fields with smallerdiscriminants. Similarly, the lattice built from Q(â2) achieves a 4.5 dBgap, still outperforming the single-level Construction A lattice from [29],which shows a 5.5 dB gap in this setting.These observations affirm that lattices constructed over number fieldswith lower discriminants not only yield denser packing (reflected in the de-terminant of the lattice) but also support better shaping and decoding be-havior under block fading. Note that the discriminant âK directly influencesthe volume of the embedded lattice, and hence its packing density. Lowerdiscriminants typically lead to smaller lattice volumes, which correspond todenser sphere packings and improved resistance to noise and fading. The use1294.5. Simulation Results10 15 20 25 30 35 40 4510-410-310-210-1100Frame Error Rate (FER)Figure 4.2: Decoding of diversity-2 Construction D lattices withm = 2, 7, 10.of layered decoding in the Construction D framework further enhances thisbenefit by allowing careful alignment of lattice layers to the fading profile.Figure 4.3 shows the decoding performance of several double-diversity(n = 2) Construction D lattices constructed over the quadratic field Q(â2),using different numbers of nested codes. All configurations are built witha fixed block length of N = 100, while the number of layersâi.e., nestedcodesâvaries from two to four. This comparison highlights the impact oflattice depth on error performance when decoding is performed using thehybrid MLâSC algorithm described in Section 4.4.1304.5. Simulation ResultsThe results reveal a clear trend: increasing the number of nested codesleads to a gradual degradation in decoding performance. This behavioris rooted in the layered structure of the decoder. The top layers are de-coded using ML decoding, which is computationally intensive but highlyreliable. However, as more layers are added, the remaining componentsmust be decoded using SC. Unlike ML, SC assumes that all previously de-coded layers are correct. If an error occurs in any early SC-decoded layer,it contaminates the input to the next layerâthrough incorrect interferencecancellationâleading to error propagation across the remaining layers. Thischain reaction becomes more likely as the number of SC-dependent layersincreases.In addition to this, each deeper layer corresponds to a sublattice withfiner granularity and smaller minimum distance, which makes it more vul-nerable to both noise and residual interference from upper layers. The lowerthe minimum distance of a lattice, the more likely decoding errors becomeunder noisy or fading conditions. Consequently, while adding more layersincreases the flexibility and granularity of the code design, it also exposesthe system to more decoding fragility unless sufficient diversity or error pro-tection is available.Quantitatively, at a target FER of 10â4, the four-layer lattice withparameters [k4, k3, k2, k1] = [90, 80, 70, 60] shows a decoding gap of ap-proximately 4.9 dB from its POL. The three-layer configuration [90, 80, 70]achieves a smaller gap of 4.3 dB, and the two-layer lattice [90, 80] outper-forms both with a 4.1 dB gap. These results indicate that, under the samediversity order and block length, a shallower construction with fewer lay-ersâparticularly one that relies more heavily on ML decodingâcan offerbetter overall performance due to reduced sensitivity to error propagation.1314.5. Simulation Results10 15 20 25 30 35 40 4510-410-310-210-1100Frame Error Rate (FER)Figure 4.3: Decoding of diversity-2 Construction D lattices over Q(â2) fora = 2, 3, 4.Figure 4.4 presents the decoding performance of Construction D latticesdesigned for triple-diversity (n = 3) Rayleigh block fading channels. All lat-tices are constructed over the same totally real cubic number fieldK = Q(ν),where ν is a root of the irreducible polynomial f(x) = x3 â x2 â 3x+ 1. Apolynomial f(x) â Q[x] is said to be irreducible over Q if it cannot be fac-tored into a product of non-constant polynomials with rational coefficients.In this case, the irreducibility of f(x) ensures that the field extension Q(ν)has degree three over Q, and hence defines a cubic number field. The dis-criminant of this field is dK = 148, a relatively small value among cubic1324.5. Simulation Results5 10 15 20 25 30 3510-410-310-210-1100Frame Error rate (FER)Figure 4.4: Decoding of diversity-3 Construction D lattices over the numberfield Q(ν) with discriminant dK = 148.fields, which contributes to favorable packing properties in the resulting lat-tices. Since K admits three real embeddings, each codeword is mappedto a three-dimensional Euclidean vector, and the resulting lattices benefitfrom both algebraic structure and full-rank diversity. By fixing the numberfield and discriminant across all configurations, this comparison isolates theimpact of code parameters and lattice dimensionality on decoding perfor-mance. Three binary Construction D lattices are evaluated, each built usingtwo nested codes of different rates and block lengths. The first lattice, de-fined by [N, k2, k1] = [100, 50, 40], results in a lattice of dimension 300. At a1334.5. Simulation ResultsFER of 10â4, it operates with a gap of approximately 5.1 dB from the POLcorresponding to diversity order three.The second configuration, also with dimension 300 but with higher-ratenested codes [100, 90, 80], shows improved performance with a reduced gapof 4.8 dB. This gain reflects the improved robustness of the ML-decodedlayers and reduced susceptibility to error propagation during the SC phase.The third configuration increases the block length to N = 500 with codeparameters [500, 250, 200], leading to a high-dimensional lattice of size 1500.Despite its larger dimension and potential shaping advantages, it exhibits awider decoding gap of 6.4 dB. This outcome highlights the trade-off betweenlattice dimension and decoder vulnerability: as dimensionality grows, morelayers are handled via successive cancellation, which compounds the risk oferror propagation under non-ideal channel conditions.These results illustrate that even when the algebraic properties and pack-ing density of the base number field are favorable, the practical performanceof Construction D lattices depends heavily on decoder structure and codeparameter choices. A concise summary of these configurations, includingtheir distance to POL at FER = 10â4, is provided in Table 4.1.1344.6. ConclusionTable 4.1: Performance results of full-diversity Construction-D andConstruction-A lattices.Lattice (K, dK) n a [N, ka, . . . , k1] FER= 10â4Construction-D (Q(â2), 8) 2 2 [100, 50, 40] 4.50 dBConstruction-D (Q(â7), 28) 2 2 [100, 50, 40] 4.80 dBConstruction-D (Q(â10), 40) 2 2 [100, 50, 40] 7.00 dBConstruction-D (Q(â7), 28) 2 2 [500, 250, 200] 8.10 dBConstruction-D (Q(â10), 40) 2 2 [500, 250, 200] 8.40 dBConstruction-D (Q(â2), 8) 2 2 [500, 250, 200] 7.90 dBConstruction-D (Q(â2), 8) 2 2 [100, 90, 80] 4.10 dBConstruction-D (Q(â2), 8) 2 3 [100, 90, 80, 70] 4.30 dBConstruction-D (Q(â2), 8) 2 4 [100, 90, 80, 70, 60] 4.90 dBConstruction-D (Q(ν), 148) 3 2 [100, 90, 80] 4.80 dBConstruction-D (Q(ν), 148) 3 2 [100, 50, 40] 5.10 dBConstruction-D (Q(ν), 148) 3 2 [500, 250, 200] 6.40 dBConstruction-A (Q(â2), 8) 2 1 [100, 50] 5.50 dBConstruction-A (Q(â7), 28) 2 1 [100, 50] 7.25 dBConstruction-A (Q(â10), 40) 2 1 [100, 50] 8.45 dBConstruction-A (Q(â10), 40) 2 1 [500, 250] 8.60 dBConstruction-A (Q(â2), 8) 2 1 [100, 90] 4.20 dBConstruction-A (Q(ν), 148) 3 1 [100, 50] 3.65 dBConstruction-A (Q(ν), 148) 3 1 [500, 250] 4.30 dB4.6 ConclusionThis chapter has explored the design and decoding of Construction-Dlattices derived from nested linear codes over prime ideals in totally realalgebraic number fields, targeting reliable communication over block-fading(BF) channels. By harnessing the algebraic structure and geometric regu-larity inherent in such lattices, we established a flexible framework capableof achieving full diversity in non-ergodic fading environments. The multi-layered nature of Construction D facilitates a modular design approach,where each nested code contributes to the overall lattice shaping, packing,1354.6. Conclusionand diversity properties.A central contribution of this work lies in the development of a decodingalgorithm that balances complexity and performance. Specifically, the pro-posed hybrid decoding strategy applies maximum-likelihood (ML) decodingto the upper layers and successive cancellation (SC) decoding to the remain-ing ones. This architecture achieves linear decoding complexity in the latticedimension while maintaining robust error performance in the most criticallayers.Simulation results confirmed that Construction-D lattices outperformtheir Construction-A counterparts in scenarios with limited diversity, par-ticularly in the two-layer, diversity-2 case. These improvements are morepronounced when lattices are built over number fields with small discrimi-nants, leading to denser packings and improved minimum distance. How-ever, the benefits of increasing the number of nested layers are not alwaysrealized in practice. As the number of layers grows, the decoder relies moreheavily on SC decoding, making the deeper layers vulnerable to error prop-agation. This phenomenon limits the attainable performance, especiallyunder moderate SNR conditions.Furthermore, the current decoding scheme does not enforce parity-checkconstraints at each level, which, while reducing complexity, constrains theeffective diversity order to n. This trade-off highlights the importance ofdecoder design in preserving the full diversity potential of the underlyinglattice.Overall, the findings of this chapter underscore the strength of Con-struction D as a structured and scalable design paradigm for lattice codingin fading environments. However, they also emphasize that achieving op-timal performance requires careful calibration of code parameters, nesting1364.6. Conclusiondepth, and decoder structure. These insights serve as the foundation forthe contributions presented in the following paper, where we formalize thisframework, provide rigorous decoding analysis, and offer extended simula-tion results.137Chapter 5ConclusionThis thesis has addressed both theoretical and practical aspects of alge-braic lattice design for communication over noisy and fading channels. Thework is structured around two central threads: the foundational theory oflattice existence and goodness (via Construction A), and the developmentand performance evaluation of multi-level algebraic lattices (via Construc-tion D) designed specifically for block fading scenarios.In the first part of the thesis, we revisited the existence of good algebraiclattices for the additive white Gaussian noise (AWGN) channel using a moreaccessible and streamlined approach. Specifically, we provided a simpli-fied proof for the existence of lattices constructed from imaginary quadraticnumber fields via Construction A. This approach relied on the arithmeticstructure of quadratic integer rings and their ideal decomposition, leadingto lattices that are provably good for both coding and quantization. Theuse of discrete dither in our construction played a key role in simplifying theproof process and sharpening our understanding of second moment proper-ties. Importantly, we demonstrated that these algebraic lattices outperformconventional integer lattices in terms of both normalized second momentand error probability at high dimensions, reinforcing their applicability topractical communication systems.In the second part of the thesis, the focus shifted to Construction D â138Chapter 5. Conclusiona multi-level lattice construction adapted to the block fading (BF) channel.A comprehensive framework was developed for constructing full-diversitylattices using nested binary linear codes and the structure of totally realnumber fields. The resulting Construction D lattices were shown to inte-grate naturally with block fading models, leveraging the diversity potentialinherent in the field embeddings.A major contribution here was the proposal of a decoding algorithm tai-lored to the multi-layer structure of Construction D lattices. This hybridmethod combines maximum-likelihood (ML) decoding at the top layers withsuccessive cancellation (SC) decoding for the remaining levels. The layeredapproach enables computational efficiency, scaling linearly with the latticedimension while still capturing diversity. Semi-systematic generator matri-ces were introduced to support practical encoding and decoding, preservingstructural clarity in implementation.Through both theoretical analysis and extensive simulations, we demon-strated that Construction D lattices provide a clear performance advantageover their Construction A counterparts in block fading environments, es-pecially in double-diversity cases with two or three layers. When the un-derlying number fields had small discriminants, the lattices achieved higherpacking density and improved robustness against fading and noise. However,our findings also revealed that increasing the number of nested layers doesnot always yield better performance. In fact, beyond a certain point, perfor-mance degraded due to error propagation in the SC decoding phase. Thiswas most notable in high-dimensional lattices with more than three levels,where the lower minimum distances of deeper sublattices led to increasedvulnerability under practical SNR conditions.This observation underlines a critical design insight: the performance of139Chapter 5. ConclusionConstruction D lattices is not solely determined by algebraic elegance or lay-ering depth, but by a careful interplay between code rates, diversity, decodercomplexity, and number field properties. In systems with limited diversity,moderate SNR, or tight complexity constraints, shallower constructions builtfrom high-rate codes and fields with small discriminants appear to offer themost favorable balance.Outlook and Future WorkWhile this thesis lays a strong foundation for the application of algebraiclattices in both AWGN and block fading channels, several directions remainopen for further research:â Decoding Enhancements: Developing advanced SC-based decoders,soft-input techniques, or list decoding mechanisms may help mitigatethe performance degradation observed in deeper Construction D lat-tices.â Alternative Number Fields: Extending the framework to non-monogenicfields, complex multiplication (CM) fields, or number fields with richerunit group structures could yield lattices with improved shaping orasymptotic behavior.â Modulation and Coding Integration: Integrating Construction D lat-tices into joint source-channel coding frameworks or lattice-based mod-ulation schemes could further increase system efficiency.â Hardware Implementation: Exploring efficient encoder and decoderarchitectures on FPGA or ASIC platforms will be essential for as-sessing the practical feasibility of deploying these lattices in real-time140Chapter 5. Conclusioncommunication systems.â Applications in Multi-User Systems: The inherent structure of Con-struction D lattices lends itself to multi-user and multiple-access sce-narios; investigating their role in such settings is a promising futuredirection.In conclusion, this thesis has shown that algebraic lattice construc-tionsâparticularly those rooted in number theory and tailored for struc-tured decodingâoffer a promising path toward robust and efficient commu-nication systems. Both theoretically grounded and practically motivated,the work advances the state of knowledge in lattice-based coding and opensnew avenues for future research and implementation.141Bibliography[1] E. Agrell and B. Allen, On the best lattice quantizers, IEEE Trans-actions on Information Theory, 69 (2023), pp. 7650â7658. â pages 34[2] S. Alaca and K. S. Williams, Introductory Algebraic Number The-ory, Cambridge University Press, 2003. â pages 32[3] N. A. Alkadri, N. Do¨ttling, and S. 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Thesis/Dissertation
eng
Electrical Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Feng, Chen
Design and analysis of lattice-based communication schemes : a study of constructions A and D
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