Codes on Algebraic Curves - Serguei A. Stepanov

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Résumé :
This book provides a self-contained introduction to the theory of error-correcting codes and related topics in number theory, Algebraic Geometry and the theory of Sphere Packings. The material is presented in an easily understandable form.br/ This book is devoted to geometric Goppa codes; the recently discovered areas which combines Coding Theory, Algebraic Geometry, Number Theory, and Theory of Sphere Packings. It has an interdisciplinary nature and demonstrates the close interconnection of Coding Theory with various classical areas of mathematics. There are four main themes in the book. The first is a brief exposition of the basic concepts and facts of error-correcting code theory. The second is a complete presentation of the theory of algebraic curves; especially the curves defined over finite fields. The third is a detailed description of the theory of elliptic and modular codes, and their reductions modulo a prime number. The fourth is a construction of geometric Gappa codes producing rather long linear codes with very good parameters coming from algebraic curves, and with a lot of rational points.br/ The aim of the book is to present these themes in a simple, easily understandable manner, and explain their close interconnection. At the same time the book introduces the reader to topics which are at the forefront of current research.

Sommaire:
I. Error-Correcting Codes.- 1 Codes and Their Parameters.- 1.1. Introduction.- 1.2. Finite Fields.- 1.3. Linear Codes.- 1.4. Spectrum and Duality.- Exercises.- 2 Bounds on Codes.- 2.1. Upper Bounds.- 2.2. The Linear Programming Bound.- 2.3. Lower Bounds.- Exercises.- 3 Examples and Constructions.- 3.1. Codes of Genus Zero.- 3.2. Some Families of Codes.- 3.3. Constructing Codes from other Codes.- Exercises.- II. Algebraic Curves and Varieties.- 4 Algebraic Curves.- 4.1. Algebraic Varieties.- 4.2. Non-Singular Curves.- 4.3. Divisors on Algebraic Curves.- 4.4. The Riemann-Roch Theorem.- 4.5. Hurwitz and Pl?cker Genus Formulas.- 4.6. Special Divisors.- Exercises.- 5 Curves over a Finite Field.- 5.1. Rational Points and Divisors.- 5.2. The Zeta-Function of a Curve.- 5.3. L-Functions of Artin.- 5.4. Algebraic Function Fields.- Exercises.- 6 Counting Points on Curves over Finite Fields.- 6.1. The Number of Rational Points on a Curve.- 6.2. Character Sums.- 6.3. Asymptotics.- Exercises.- III. Elliptic and Modular Curves.- 7 Elliptic Curves.- 7.1. The Group Law.- 7.2. The j-Invariant.- 7.3. Isogenies.- 7.4. Elliptic Curves over Finite Fields.- 7.5. Elliptic Functions.- Exercises.- 8 Classical Modular Curves.- 8.1. Congruence Subgroups.- 8.2. The Curves X(N), X0(N), and X1(N).- 8.3. Hecke Operators.- 8.4. The Petersson Inner Product.- Exercises.- 9 Reductions of Modular Curves.- 9.1. Reductions and Moduli Spaces.- 9.2. The Igusa Theorem.- 9.3. The Eichler-Shimura Congruence Relation.- 9.4. The Eichler-Selberg Trace Formula.- Exercises.- IV. Geometric Goppa Codes.- 10 Constructions and Properties.- 10.1. L-Construction.- 10.2. ?-Construction.- 10.3. Parameters.- 10.4. Duality and Spectra.- Exercises.- 11 Examples.- 11.1. Codes of Small Genera.- 11.2. Elliptic and Hermitian Codes.- 11.3. Codes on Fiber Products.- 11.4. Codes on Classical Modular Curves.- 11.5. Codes on Artin-Schreier Coverings.- 11.6. Codes on Trace-Norm Curves.- Exercises.- 12 Decoding Geometric Goppa Codes.- 12.1. The Decoding Problem.- 12.2. The Basic and Modified Algorithms.- 12.3. An Improvement of the Modified Algorithm.- 12.4. Majority Voting for Unknown Syndromes.- 12.5. Faster Decoding.- Exercises.- 13 Bounds.- 13.1. Asymptotic Bounds.- 13.2. Constructive Bounds.- 13.3. Other Bounds.- Exercises.- List of Notations.