Rational points on curves over finite fields theory and applications - Niederreiter, Harald (Cambridge University Press, 2001)
Includes bibliographical references (p. 227-239) and index
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