Algebraic Function Fields and Codes
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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algebraic extension algebraic function ﬁelds algebraic geometry algebraically closed assume automorphism Bound canonical divisor char Choose closure coding theory consider constant field extension Corollary cyclic defined Deﬁnition degA degP denote different exponent Div(F divisor class equation exists F/Wq field F field F/K ﬁnite following hold full constant field function field G K[x Galois extension genus g Goppa codes Hasse-Weil hence implies integral integral closure irreducible isomorphism Lemma Let F Let F/K linear minimal polynomial minimum distance notation obtain pairwise distinct place of F/K places of degree pole prime element Proposition prove rational function rational function field rational places resp Riemann-Roch Theorem satisfies separable extension sequence Show splits completely subfield subgroup Suppose tower Q Triangle Inequality unique unramified valuation ring vector space vp(u vp(x vp(z weakly ramified zero