Algebraic Function Fields and Codes

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Springer Science & Business Media, 2009 - Computers - 355 pages
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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
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