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Abstract

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.