Title Rational points on curves over finite fields : theory and applications / Harald Niederreiter, Chaoping Xing

Published Cambridge ; New York : Cambridge University Press, 2001

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Description 1 online resource (x, 245 pages) Series London Mathematical Society lecture note series ; 285 London Mathematical Society lecture note series ; 285. Contents Background on Function Fields -- Riemann-Roch Theorem -- Divisor Class Groups and Ideal Class Groups -- Algebraic Extensions and the Hurwitz Formula -- Ramification Theory of Galois Extensions -- Constant Field Extensions -- Zeta Functions and Rational Places -- Class Field Theory -- Local Fields -- Newton Polygons -- Ramification Groups and Conductors -- Global Fields -- Ray Class Fields and Hilbert Class Fields -- Narrow Ray Class Fields -- Class Field Towers -- Explicit Function Fields -- Kummer and Artin-Schreier Extensions -- Cyclotomic Function Fields -- Drinfeld Modules of Rank 1 -- Function Fields with Many Rational Places -- Function Fields from Hilbert Class Fields -- Function Fields from Narrow Ray Class Fields -- The First Construction -- The Second Construction -- The Third Construction -- Function Fields from Cyclotomic Fields -- Explicit Function Fields -- Asymptotic Results -- Asymptotic Behavior of Towers -- The Lower Bound of Serre -- Further Lower Bounds for A(q[superscript m]) -- Explicit Towers -- Lower Bounds on A(2), A(3), and A(5) -- Applications to Algebraic Coding Theory -- Goppa's Algebraic-Geometry Codes -- Beating the Asymptotic Gilbert-Varshamov Bound -- NXL Codes -- XNL Codes -- A Propagation Rule for Linear Codes -- Applications to Cryptography -- Background on Stream Ciphers and Linear Complexity -- Constructions of Almost Perfect Sequences -- A Construction of Perfect Hash Families -- Hash Families and Authentication Schemes -- Applications to Low-Discrepancy Sequences Summary Discussion of theory and applications of algebraic curves over finite fields with many rational points Bibliography Includes bibliographical references (pages 227-239) and index Notes English Print version record Subject Curves, Algebraic. Finite fields (Algebra) Rational points (Geometry) Coding theory. MATHEMATICS -- Geometry -- Algebraic. Coding theory Curves, Algebraic Finite fields (Algebra) Rational points (Geometry) Algebraische Kurve Algebraische Codierung Galois-Feld Rationaler Punkt Eindige lichamen. Krommen. Corpos de classe. Teoria dos números. Criptologia. Computabilidade e complexidade. Courbes algébriques. Champs modulaires. Form Electronic book Author Xing, Chaoping, 1963- ISBN 9781107089297 1107089298 9781107325951 1107325951 9781107095540 1107095549 1299707203 9781299707207

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