Codes on Algebraic Curves - Serguei A. Stepanov

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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.