Rational points on curves over finite fields : theory and applications

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Bibliographic Information

Rational points on curves over finite fields : theory and applications

Harald Niederreiter, Chaoping Xing

（London Mathematical Society lecture note series, 288[i.e.285]）

Cambridge University Press, 2001

  • : pbk

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Note

Bibliography: p. 227-239

Includes index

"London Mathematical Society lecture note series. 288"

Description and Table of Contents

Description

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 over finite fields with many rational points (relative to the genus). Recently, the authors discovered another important application of such curves, namely to the construction of low-discrepancy sequences. These sequences are needed for numerical methods in areas as diverse as computational physics and mathematical finance. This has given additional impetus to the theory of, and the search for, algebraic curves over finite fields with many rational points. This book aims to sum up the theoretical work on algebraic curves over finite fields with many rational points and to discuss the applications of such curves to algebraic coding theory and the construction of low-discrepancy sequences.

Table of Contents

  • 1. Background on function fields
  • 2. Class field theory
  • 3. Explicit function fields
  • 4. Function fields with many rational places
  • 5. Asymptotic results
  • 6. Applications to algebraic coding theory
  • 7. Applications to cryptography
  • 8. Applications to low-discrepancy sequences.

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