Algebraic Function Fields and Codes / Edition 2

by Henning Stichtenoth

Overview

15 years after the first 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 difiers from the first 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 fields overfinite fields, 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 fields to coding theory. The codes which are constructed from algebraic function fields were first introduced by V. D. Goppa. Accordingly I referred to them in the first 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.

Product Details

ISBN-13:	9783642095566
Publisher:	Springer Berlin Heidelberg
Publication date:	11/19/2010
Series:	Graduate Texts in Mathematics , #254
Edition description:	Softcover reprint of hardcover 2nd ed. 2008
Pages:	360
Product dimensions:	6.10(w) x 9.25(h) x 0.03(d)

Table of Contents

Foundations of the Theory of Algebraic Function Fields.- Algebraic Geometry Codes.- Extensions of Algebraic Function Fields.- Differentials of Algebraic Function Fields.- Algebraic Function Fields over Finite Constant Fields.- Examples of Algebraic Function Fields.- Asymptotic Bounds for the Number of Rational Places.- More about Algebraic Geometry Codes.- Subfield Subcodes and Trace Codes.

Editorial Reviews

The text has two objectives, the first of which is to cover algebraic function fields. Emphasis is on function fields over a finite constant field, leading into zeta functions and the Hasse-Weil theorem. Error correcting codes, used for the reliable transmission of information, are based on the mathematical construction by V.D. Goppa. Algebraic function fields provide the most elementary approach to Goppa's ideas, so the second objective is to give an introduction to Goppa's algebraic-geometric codes. Numerous examples are included. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Booknews

From the reviews of the second edition:

"In this book we have an exposition of the theory of function fields in one variable from the algebraic point of view … . The book is carefully written, the concepts are well motivated and plenty of examples help to understand the ideas and proofs and so it can be used as a textbook for an introductory course on the (classical) arithmetic of function fields with an application to coding theory." (Felipe Zaldivar, MAA Online, January, 2009)

From the Publisher