Skip to main content
SpringerLink
Log in

Codes on Algebraic Curves

  • Book
  • © 1999

Overview

Authors:
  1. Serguei A. Stepanov

    1. Bilkent University, Ankara, Turkey
      Steklov Mathematical Institute, Moscow, Russia

    View author publications

    You can also search for this author in PubMed   Google Scholar

This is a preview of subscription content, log in via an institution to check access.

Access this book

Log in via an institution
eBook USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Other ways to access

Licence this eBook for your library

Institutional subscriptions

Table of contents (13 chapters)

  1. Front Matter

    Pages i-xiii
    Download chapter PDF

  2. Error-Correcting Codes

    1. Front Matter

      Pages 1-1
      Download chapter PDF

    2. Codes and Their Parameters

      • Serguei A. Stepanov
      Pages 3-23

    3. Bounds on Codes

      • Serguei A. Stepanov
      Pages 25-39

    4. Examples and Constructions

      • Serguei A. Stepanov
      Pages 41-67

  3. Algebraic Curves and Varieties

    1. Front Matter

      Pages 69-69
      Download chapter PDF

    2. Algebraic Curves

      • Serguei A. Stepanov
      Pages 71-101

    3. Curves over a Finite Field

      • Serguei A. Stepanov
      Pages 103-142

    4. Counting Points on Curves over Finite Fields

      • Serguei A. Stepanov
      Pages 143-172

  4. Elliptic and Modular Curves

    1. Front Matter

      Pages 173-173
      Download chapter PDF

    2. Elliptic Curves

      • Serguei A. Stepanov
      Pages 175-192

    3. Classical Modular Curves

      • Serguei A. Stepanov
      Pages 193-217

    4. Reductions of Modular Curves

      • Serguei A. Stepanov
      Pages 219-240

  5. Geometric Goppa Codes

    1. Front Matter

      Pages 241-241
      Download chapter PDF

    2. Constructions and Properties

      • Serguei A. Stepanov
      Pages 243-255

    3. Examples

      • Serguei A. Stepanov
      Pages 257-288

    4. Decoding Geometric Goppa Codes

      • Serguei A. Stepanov
      Pages 289-314

    5. Bounds

      • Serguei A. Stepanov
      Pages 315-322

  6. Back Matter

    Pages 323-350
    Download chapter PDF
Back to top

Keywords

About this book

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.

Authors and Affiliations

  • Bilkent University, Ankara, Turkey

    Serguei A. Stepanov

  • Steklov Mathematical Institute, Moscow, Russia

    Serguei A. Stepanov

Bibliographic Information

  • Book Title: Codes on Algebraic Curves

  • Authors: Serguei A. Stepanov

  • DOI: https://doi.org/10.1007/978-1-4615-4785-3

  • Publisher: Springer New York, NY

  • eBook Packages: Springer Book Archive

  • Copyright Information: Kluwer Academic/Plenum Publishers 1999

  • Hardcover ISBN: 978-0-306-46144-6Published: 31 July 1999

  • Softcover ISBN: 978-1-4613-7167-0Published: 21 October 2012

  • eBook ISBN: 978-1-4615-4785-3Published: 06 December 2012

  • Edition Number: 1

  • Number of Pages: XIII, 350

  • Topics: Algebra, Electrical Engineering, Algebraic Geometry, Algorithms

Publish with us

Policies and ethics

Back to top

Search

Navigation