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Algebraic function fields and codes. 2nd ed. (English) Zbl 1155.14022
Graduate Texts in Mathematics 254. Berlin: Springer (ISBN 978-3-540-76877-7/hbk). xiii, 355 p. (2009).
This is the second edition of an already classical book whose first edition appeared in 1993 [Universitext. Berlin: Springer-Verlag (1993; Zbl 0816.14011)]. It covers the two interrelated topics of algebraic functions fields of one variable (or what is equivalent the theory of algebraic curves) and algebraic-geometry (AG) codes.
On one hand the book gives a purely algebraic exposition of the theory of function fields over a (perfect) field. Chapter 1 gives the basic concepts and proves the Riemann-Roch theorem, Chapter 3 studies algebraic extensions of function fields, Chapter 4 develops the theory of differentials and Chapter 6 shows some particular examples such as elliptic and hyperelliptic function fields.
With the mind in applications to coding theory Chapter 5 considers the particular case of finite constant field, giving the proof of the Hasse-Weil theorem, and an entirely new Chapter (the 7) has been added in the present edition, devoted to the asymptotic theory of function fields over a finite field.
The theory of AG codes (called geometric Goppa codes in the first edition) is the subject of the remaining Chapters. Chapter 2 gives a brief introduction to AG codes while Chapter 8 contains some more advanced topics such as the Tsfasman-Vladut-Zink bound or the Skorobogatov-Vladut decoding algorithm for AG-codes. Finally Chapter 9 approaches the study of subfield subcodes and trace codes and their relations with functions fields.
The present edition also gets richer with the inclusion of a list of proposed exercises (of different levels of difficulty) at the end of each chapter.

MSC:
14H05 Algebraic functions and function fields in algebraic geometry
14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
94-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to information and communication theory
94B27 Geometric methods (including applications of algebraic geometry) applied to coding theory
11R58 Arithmetic theory of algebraic function fields
11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
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