Algebraic Function Fields and Codes15 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. |
Contents
Foundations of the Theory of Algebraic Function Fields | 1 |
Algebraic Geometry Codes | 45 |
Extensions of Algebraic Function Fields 67 | 66 |
Differentials of Algebraic Function Fields | 155 |
Algebraic Function Fields over Finite Constant Fields | 185 |
Examples of Algebraic Function Fields | 217 |
Asymptotic Bounds for the Number of Rational Places 243 | 242 |
More about Algebraic Geometry Codes | 289 |
Subfield Subcodes and Trace Codes 311 | 310 |
Appendix A Field Theory | 327 |
Appendix B Algebraic Curves and Function Fields | 335 |
List of Notations | 345 |
References | 349 |
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Common terms and phrases
algebraic function fields assertion assume automorphism basis Bound called Chapter char Choose claim closed closure codes completely conclude consequence consider constant field contains convergent Corollary corresponding cyclic defined Definition denote determine differential distance distinct Div(F divisor easily element equation equivalent Example exercise exists fact field field extension finite follows full constant function field Galois genus g given hence holds ideal immediately implies Inequality integral irreducible isomorphism Lemma linear notation Note Observe obtain obvious particular place of F/K place P G pole polynomial prime principal Proof properties Proposition prove ramified rational function field Remark residue class resp result ring roots satisfies separable sequence Show space splits step subgroup Suppose Theorem theory tower unique valuation vector write yields zero