Algebraic-Geometric Codes / Edition 1

by M. Tsfasman, S.G. Vladut

Overview

'Et moi, ..., si j'avait su comment en revenir, One service mathematics has rendered the je n'y serais point aIle.' human race. It has put common sense back Jules Verne where it belongs, on the topmost shelf next to the dusty canister labelled 'discarded non- sense'. The series is divergent; therefore we may be able to do something with it. Eric T. Bell O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and non- linearities abound. Similarly, all kinds of parts of mathematics serve as tools for other parts and for other sciences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One service topology has rendered mathematical physics .. .'; 'One service logic has rendered com- puter science .. .'; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way form part of the raison d' etre of this series.

Product Details

ISBN-13:	9780792307273
Publisher:	Springer Netherlands
Publication date:	04/30/1991
Series:	Mathematics and its Applications , #58
Edition description:	1991
Pages:	667
Product dimensions:	6.10(w) x 9.25(h) x 0.36(d)

Table of Contents

1. Codes.- 1.1. Codes and their parameters.- 1.2. Examples and constructions.- 1.3. Asymptotic problems.- 2. Curves.- 2.1. Algebraic curves.- 2.2. Riemann-Roch theorem.- 2.3. Rational points.- 2.4. Elliptic curves.- 2.5. Singular curves.- 2.6. Reductions and schemes.- 3. AG-Codes.- 3.1. Constructions and properties.- 3.2. Examples.- 3.3. Decoding.- 3.4. Asymptotic results.- 4. Modular Codes.- 4.1. Codes on classical modular curves.- 4.2. Codes on Drinfeld curves.- 4.3. Polynomiality.- 5. Sphere Packings.- 5.1. Definitions and examples.- 5.2. Asymptotically dense packings.- 5.3. Number fields.- 5.4. Analogues of AG-codes.- Appendix. Summary of results and tables.- A.1. Codes of finite length.- A.1.1. Bounds.- A.1.2. Parameters of certain codes.- A.1.3. Parameters of certain constructions.- A.1.4. Binary codes from AG-codes.- A.2. Asymptotic bounds.- A.2.1. List of bounds.- A.2.2. Diagrams of comparison.- A.2.3. Behaviour at the ends.- A.2.4. Numerical values.- A.3. Additional bounds.- A.3.1. Constant weight codes.- A.3.2. Self-dual codes.- A.4. Sphere packings.- A.4.1. Small dimensions.- A.4.2. Certain families.- A.4.3. Asymptotic results.- Author index.- List of symbols.

Editorial Reviews

Two of the leading mathematicians in the field present a comprehensive treatment of the recently emergent field which combines algebraic geometry and coding theory. Written for graduate students and researchers, the text begins with essentially first principles, and leads to the most recent published and unpublished results. Problems and hints for their solutions are included, as well as many open problems. The appendix contains equations and tables of some asymptotic bounds, tables of parameters of certain families of codes, and tables of dense sphere packings. Doesn't assume any initial knowledge either of coding theory or of algebraic geometry. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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