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.

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.

Reviewer: Juan Tena Ayuso (Valladolid)

##### 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) |