Item description for Algebraic Function Fields and Codes by Henning Stichtenoth...
This book has two objectives. The first is to fill a void in the existing mathematical literature by providing a modern, self-contained and in-depth exposition of the theory of algebraic function fields. Topics include the Riemann-Roch theorem, algebraic extensions of function fields, ramifications theory and differentials. Particular emphasis is placed on function fields over a finite constant field, leading into zeta functions and the Hasse-Weil theorem. Numerous examples illustrate the general theory. Error-correcting codes are in widespread use for the reliable transmission of information. Perhaps the most fascinating of all the ties that link the theory of these codes to mathematics is the construction by V.D. Goppa, of powerful codes using techniques borrowed from algebraic geometry. Algebraic function fields provide the most elementary approach to Goppa's ideas, and the second objective of this book is to provide an introduction to Goppa's algebraic-geometric codes along these lines. The codes, their parameters and links with traditional codes such as classical Goppa, Peed-Solomon and BCH codes are treated at an early stage of the book. Subsequent chapters include a decoding algorithm for these codes as well as a discussion of their subfield subcodes and trace codes. Stichtenoth's book will be very useful to students and researchers in algebraic geometry and coding theory and to computer scientists and engineers interested in information transmission.
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Est. Packaging Dimensions: Length: 9.22" Width: 6.36" Height: 0.69" Weight: 0.9 lbs.
Release Date Jun 25, 1993
ISBN 3540564896 ISBN13 9783540564898
Reviews - What do customers think about Algebraic Function Fields and Codes?
an introduction to Goppa codes Aug 22, 2003
Coding theory is fundamental to make digital transmission technology work efficiently and usually uses Reed-Solomon codes. The "natural" extension of those codes is to consider riemann surfaces over finite fields.
The theory is developped from scratch and does not assume any knowledge of algebraic geometry. The author gave a proof of the Hasse-Weil bounds using the Zeta function. In parallel the theory of linear codes and Goppa codes is introduced from the beginning.
While the author do not consider the geometry of riemann surfaces, having a knowledge of riemann surfaces over C can help a lot. This short book should be considered as a very nice introduction to geometric goppa codes.