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Codes and curves

Author: Judy L Walker
Publisher: Providence : American Mathematical Society, June 2000.
Series: Student mathematical library, v. 7.
Edition/Format:   eBook : Document : EnglishView all editions and formats
Summary:
When information is transmitted, errors are likely to occur. Coding theory examines efficient ways of packaging data so that these errors can be detected, or even corrected. The traditional tools of coding theory have come from combinatorics and group theory. Lately, however, coding theorists have added techniques from algebraic geometry to their toolboxes. In particular, by re-interpreting the Reed-Solomon codes,  Read more...
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Genre/Form: Electronic books
Additional Physical Format: Print version:
Walker, Judy L., 1969-
Codes and curves.
Providence, RI : American Mathematical Society, ©2000
(DLC) 00038112
(OCoLC)43859497
Material Type: Document, Internet resource
Document Type: Internet Resource, Computer File
All Authors / Contributors: Judy L Walker
ISBN: 9781470418236 1470418231 082182628X 9780821826287
OCLC Number: 907358829
Target Audience: College Audience
Description: 1 online resource.
Contents: Chapter 1. Introduction to coding theory Chapter 2. Bounds on codes Chapter 3. Algebraic curves Chapter 4. Nonsingularity and the genus Chapter 5. Points, functions, and divisors on curves Chapter 6. Algebraic geometry codes Chapter 7. Good codes from algebraic geometry Appendix A. Abstract algebra review Appendix B. Finite fields Appendix C. Projects.
Series Title: Student mathematical library, v. 7.
Responsibility: Judy L. Walker.

Abstract:

When information is transmitted, errors are likely to occur. Coding theory examines efficient ways of packaging data so that these errors can be detected, or even corrected. The traditional tools of coding theory have come from combinatorics and group theory. Lately, however, coding theorists have added techniques from algebraic geometry to their toolboxes. In particular, by re-interpreting the Reed-Solomon codes, one can see how to define new codes based on divisors on algebraic curves. For instance, using modular curves over finite fields, Tsfasman, Vladut, and Zink showed that one can define a sequence of codes with asymptotically better parameters than any previously known codes. This monograph is based on a series of lectures the author gave as part of the IAS/PCMI program on arithmetic algebraic geometry. Here, the reader is introduced to the exciting field of algebraic geometric coding theory. Presenting the material in the same conversational tone of the lectures, the author covers linear codes, including cyclic codes, and both bounds and asymptotic bounds on the parameters of codes. Algebraic geometry is introduced, with particular attention given to projective curves, rational functions and divisors. The construction of algebraic geometric codes is given, and the Tsfasman-Vladut-Zink result mentioned above is discussed.

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