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

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Ever since the seminal work of Goppa on algebraic-geometry codes, rational points on algebraic curves over finite fields have been an important research topic for algebraic geometers and coding theorists. The focus in this application of algebraic geometry to coding theory is on algebraic curves ove...

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Bibliographic Details
Main Authors: Niederreiter, Harald, 1944- (Author)
Xing, Chaoping, 1963- (Author)
Language:English
Published: Cambridge : Cambridge University Press, 2001.
Series:London Mathematical Society lecture note series ; 285.
Subjects:
Curves, Algebraic.
Finite fields (Algebra)
Rational points (Geometry)
Coding theory.
Online Access:
Connect to online resource - MSU authorized users (Digital Object Identifier Permalink)
Physical Description:1 online resource (x, 245 pages) : digital, PDF file(s).
Format: Electronic eBook
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.