Algebraic-Geometric Codes / Edition 1

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Editorial Reviews

Booknews
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students and researchers, the text begins with essentially first principles, and leads to the most recent published and unpublished results.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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Product Details

  • ISBN-13: 9780792307273
  • Publisher: Springer Netherlands
  • Publication date: 4/15/1991
  • Series: Mathematics and its Applications Series , #58
  • Edition description: 1991
  • Edition number: 1
  • Pages: 667
  • Product dimensions: 1.63 (w) x 6.14 (h) x 9.21 (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.

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