# Algebraic Function Fields and Codes

Springer Science & Business Media, Feb 11, 2009 - Mathematics - 360 pages
15 years after the ?rst printing of Algebraic Function Fields and Codes,the mathematics editors of Springer Verlag encouraged me to revise and extend the book. Besides numerous minor corrections and amendments, the second edition di?ers from the ?rst one in two respects. Firstly I have included a series of exercises at the end of each chapter. Some of these exercises are fairly easy and should help the reader to understand the basic concepts, others are more advanced and cover additional material. Secondly a new chapter titled “Asymptotic Bounds for the Number of Rational Places” has been added. This chapter contains a detailed presentation of the asymptotic theory of function ?elds over ?nite ?elds, including the explicit construction of some asymptotically good and optimal towers. Based on these towers, a complete and self-contained proof of the Tsfasman-Vladut-Zink Theorem is given. This theorem is perhaps the most beautiful application of function ?elds to coding theory. The codes which are constructed from algebraic function ?elds were ?rst introduced by V. D. Goppa. Accordingly I referred to them in the ?rst edition as geometric Goppa codes. Since this terminology has not generally been - cepted in the literature, I now use the more common term algebraic geometry codes or AG codes. I would like to thank Alp Bassa, Arnaldo Garcia, Cem Guneri, ̈ Sevan Harput and Alev Topuzo? glu for their help in preparing the second edition.

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### Contents

 Foundations of the Theory of Algebraic Function Fields 1 Algebraic Geometry Codes 45 Extensions of Algebraic Function Fields 67 Differentials of Algebraic Function Fields 155 Algebraic Function Fields over Finite Constant Fields 185 Examples of Algebraic Function Fields 217 216 Asymptotic Bounds for the Number of Rational Places 243
 More about Algebraic Geometry Codes 289 288 Subﬁeld Subcodes and Trace Codes 311 Appendix A Field Theory 327 Appendix B Algebraic Curves and Function Fields 335 List of Notations 345 References 349 348 Copyright