Overview

This book has two objectives. The first is to fill a void inthe existing mathematical literature by providing a modern,self-contained and in-depth exposition of the theory ofalgebraic function fields. Topics include the Riemann-Rochtheorem, algebraic extensions of function fields,ramifications theory and differentials. Particular emphasisis placed on function fields over a finite constant field,leading into zeta functins and the Hasse-Weil theorem.Numerous examples illustrate the general theory.Error-correcting codes are in widespread use for thereliable transmission of information. Perhaps the mostfascinating of all the ties that link the theory of thesecodes to mathematics is the construction by V.D. Goppa, ofpowerful codes using techniques borrowed fromalgebraicgeometry. Algebraic function fields provide the mostelementary approach to Goppa's ideas, and the secondobjective of this book is to provide an introduction toGoppa's algebraic-geometric codesalong these lines. Thecodes, their parameters and links with traditional codessuch as classical Goppa, Peed-Solomon and BCH codes aretreated atan early stage of the book. Subsequent chaptersinclude a decoding algorithmfor these codes as well as adiscussion of their subfield subcodes and tracecodes.Stichtenoth's book will be very useful to students andresearchers in algebraic geometry and coding theory and tocomputer scientists and engineers interested in informationtransmission.

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9783540564898

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