Advertisement

Codes on Algebraic Curves

  • Serguei A. Stepanov
Book

Table of contents

  1. Front Matter
    Pages i-xiii
  2. Error-Correcting Codes

    1. Front Matter
      Pages 1-1
    2. Serguei A. Stepanov
      Pages 3-23
    3. Serguei A. Stepanov
      Pages 25-39
    4. Serguei A. Stepanov
      Pages 41-67
  3. Algebraic Curves and Varieties

    1. Front Matter
      Pages 69-69
    2. Serguei A. Stepanov
      Pages 71-101
    3. Serguei A. Stepanov
      Pages 103-142
    4. Serguei A. Stepanov
      Pages 143-172
  4. Elliptic and Modular Curves

    1. Front Matter
      Pages 173-173
    2. Serguei A. Stepanov
      Pages 175-192
    3. Serguei A. Stepanov
      Pages 193-217
    4. Serguei A. Stepanov
      Pages 219-240
  5. Geometric Goppa Codes

    1. Front Matter
      Pages 241-241
    2. Serguei A. Stepanov
      Pages 243-255
    3. Serguei A. Stepanov
      Pages 257-288
    4. Serguei A. Stepanov
      Pages 289-314
    5. Serguei A. Stepanov
      Pages 315-322
  6. Back Matter
    Pages 323-350

About this book

Introduction

This is a self-contained introduction to algebraic curves over finite fields and geometric Goppa codes. There are four main divisions in the book. The first is a brief exposition of basic concepts and facts of the theory of error-correcting codes (Part I). The second is a complete presentation of the theory of algebraic curves, especially the curves defined over finite fields (Part II). The third is a detailed description of the theory of classical modular curves and their reduction modulo a prime number (Part III). The fourth (and basic) is the construction of geometric Goppa codes and the production of asymptotically good linear codes coming from algebraic curves over finite fields (Part IV). The theory of geometric Goppa codes is a fascinating topic where two extremes meet: the highly abstract and deep theory of algebraic (specifically modular) curves over finite fields and the very concrete problems in the engineering of information transmission. At the present time there are two essentially different ways to produce asymptotically good codes coming from algebraic curves over a finite field with an extremely large number of rational points. The first way, developed by M. A. Tsfasman, S. G. Vladut and Th. Zink [210], is rather difficult and assumes a serious acquaintance with the theory of modular curves and their reduction modulo a prime number. The second way, proposed recently by A.

Keywords

Prime Prime number algebra algebraic curve algebraic geometry algebraic varieties algorithms coding theory finite field geometry interconnect material mathematics number theory programming

Authors and affiliations

  • Serguei A. Stepanov
    • 1
    • 2
  1. 1.Bilkent UniversityAnkaraTurkey
  2. 2.Steklov Mathematical InstituteMoscowRussia

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4615-4785-3
  • Copyright Information Kluwer Academic/Plenum Publishers 1999
  • Publisher Name Springer, Boston, MA
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4613-7167-0
  • Online ISBN 978-1-4615-4785-3
  • Buy this book on publisher's site