# Algebraic Function Fields and Codes

Springer Science & Business Media, Nov 20, 2008 - 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

 I 1 III 8 IV 12 V 15 VI 24 VII 31 VIII 37 IX 40
 XXXII 197 XXXIII 208 XXXIV 212 XXXV 217 XXXVI 224 XXXVII 227 XXXVIII 232 XXXIX 235

 X 45 XII 48 XIII 55 XIV 63 XV 66 XVI 68 XVII 77 XVIII 81 XIX 90 XX 112 XXI 120 XXII 130 XXIII 137 XXIV 143 XXV 145 XXVI 150 XXVII 155 XXVIII 161 XXIX 170 XXX 179 XXXI 185
 XL 242 XLI 246 XLII 259 XLIII 261 XLIV 285 XLV 289 XLVI 290 XLVII 292 XLVIII 297 XLIX 303 L 308 LI 310 LII 315 LIII 325 LIV 327 LVI 335 LVII 345 LVIII 349 LIX 352 Copyright