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Coding theory and algebraic curves Edwards, Brandon Gary


In 1981 V.D. Goppa [9] used the evaluation of rational functions on algebraic curves to define a new and exciting class of error correcting codes now known as geometric Goppa codes. As with any other error correcting code however, this construction procedure alone was only the beginning of what was need in order that these codes could be used in practice. Procedures needed to be found to aid in both the explicit construction of certain geometric Goppa codes and the creation of efficient decoding algorithms. The problems surrounding these concerns have since motivated many coding theorists to become actively involved in the study of algebraic curves and have equally sparked the interests of algebraic geometers. In this paper I introduce the topic of coding theory and highlight the successes and failures that have occurred in the attempt to bring geometric Goppa codes to their full stage of implementation. In the first two chapters, I give a basic introduction to the theory of error correcting codes, assuming no previous knowledge of the subject. Chapter 3 is concerned with the construction of rational geometric Goppa codes for which no knowledge of algebraic curves is necessary. Various positive and negative aspects of these codes are discussed which motivates the introduction of the general class of geometric Goppa codes in the following two chapters. For the material in Chapters 4 and 5, I will assume that the reader is familiar with divisors and linear systems on non-singular projective curves. After I define the class of geometric Goppa codes, I go on to discuss the advances that have been made in the search for explicit constructions of some of the best codes in this class. Finally, I present some basic decoding algorithms for both rational and general geometric Goppa codes. These algorithms demonstrate how easy it is to develop efficient decoders in both of these cases.

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