Rational points on curves over finite fields theory and applications /
Harald Niederreiter, Chaoping Xing.
- Cambridge ; New York : Cambridge University Press, (c)2001.
- 1 online resource (x, 245 pages)
- London Mathematical Society lecture note series ; 285 .
Includes bibliographies 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.