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Codes on Algebraic Curves

Codes on Algebraic Curves


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About the Book

This text provides an introduction to the theory of error-correcting codes and related topics in number theory, algebraic geometry and the theory of sphere packings. The material is presented in an easily understandable form. This book is devoted to geometric Goppa codes; the recently discovered areas which combines coding theory, algebraic geometry, number theory, and theory of sphere packings. It has an interdisciplinary nature and demonstrates the close interconnection of coding theory with various classical areas of mathematics. There are four main themes in the book, the first being a brief exposition of the basic concepts and facts of error-correcting code theory. The second is a complete presentation of the theory of algebraic curves; especially the curves defined over finite fields. The third is a detailed description of the theory of elliptic and modular codes, and their reductions modulo a prime number. The fourth is a construction of geometric Gappa codes producing rather long linear codes with very good parameters coming from algebraic curves, and with a lot of rational points.

Table of Contents:
I. Error-Correcting Codes.- 1 Codes and Their Parameters.- 2 Bounds on Codes.- 3 Examples and Constructions.- II. Algebraic Curves and Varieties.- 4 Algebraic Curves.- 5 Curves over a Finite Field.- 6 Counting Points on Curves over Finite Fields.- III. Elliptic and Modular Curves.- 7 Elliptic Curves.- 8 Classical Modular Curves.- 9 Reductions of Modular Curves.- IV. Geometric Goppa Codes.- 10 Constructions and Properties.- 11 Examples.- 12 Decoding Geometric Goppa Codes.- 13 Bounds.- List of Notations.


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Product Details
  • ISBN-13: 9780306461446
  • Publisher: Springer Science+Business Media
  • Publisher Imprint: Kluwer Academic/Plenum Publishers
  • Height: 229 mm
  • No of Pages: 350
  • Width: 152 mm
  • ISBN-10: 0306461447
  • Publisher Date: 31 Jul 1999
  • Binding: Hardback
  • Language: English
  • Returnable: Y


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