Rational Points on Curves over Finite Fields: Theory and Applications (London Mathematical Society Lecture Note Series, Series Number 285)

Rational points on algebraic curves over finite fields is a key topic for algebraic geometers and coding theorists. Here, the authors relate an important application of such curves, namely, to the construction of low-discrepancy sequences, needed for numerical methods in diverse areas. They sum up the theoretical work on algebraic curves over finite fields with many rational points and discuss the applications of such curves to algebraic coding theory and the construction of low-discrepancy sequences. 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.

Discussion of theory and applications of algebraic curves over finite fields with many rational points.

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Ever since Hasse and Weil's work in the 1930s and 1940s, algebraic curves over finite fields and their function fields have attracted both number theorists and specialists in geometry. Niederreiter and Xing, both at the National U. of Singapore, present classic field theory, recent research using the methods of algebraic geometry, and applications in algebraic coding theory and cryptography, among other topics. Includes tables of upper and lower bounds, and 188 references. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Chapter 3 Explicit Function Fields3.1 Kummer and Artin-Schreier Extensions; 3.2 Cyclotomic Function Fields; 3.3 Drinfeld Modules of Rank 1; Chapter 4 Function Rational Fields with Many Places; 4.1 Function Fields from Hilbert Class Fields; 4.2 Function Fields from Narrow Ray Class Fields; 4.2.1 The First Construction; 4.2.2 The Second Construction; 4.2.3 The Third Construction; 4.3 Function Fields from Cyclotomic Fields; 4.4 Explicit Function Fields; 4.5 Tables; Chapter 5 Asymptotic Results; 5.1 Asymptotic Behavior of Towers; 5.2 The Lower Bound of Serre; 5.3 Further Lower Bounds for A(qm) Cover; Series Page; Title; Copyright; Dedication; Contents; Preface; Chapter 1 Background on Function Fields; 1.1 Riemann-Roch Theorem; 1.2 Divisor Class Groups and Ideal Class Groups; 1.3 Algebraic Extensions and the Hurwitz Formula; 1.4 Ramification Theory of Galois Extensions; 1.5 Constant Field Extensions; 1.6 Zeta Functions and Rational Places; Chapter 2 Class Field Theory; 2.1 Local Fields; 2.2 Newton Polygons; 2.3 Ramification Groups and Conductors; 2.4 Global Fields; 2.5 Ray Class Fields and Hilbert Class Fields; 2.6 Narrow Ray Class Fields; 2.7 Class Field Towers 5.4 Explicit Towers5.5 Lower Bounds on A(2), A(3), and A(5); Chapter 6 Applications to Algebraic Coding Theory; 6.1 Goppa's Algebraic-Geometry Codes; 6.2 Beating the Asymptotic Gilbert-Varshamov Bound; 6.3 NXL Codes; 6.4 XNL Codes; 6.5 A Propagation Rule for Linear Codes; Chapter 7 Applications to Cryptography; 7.1 Background on Stream Ciphers and Linear Complexity; 7.2 Constructions of Almost Perfect Sequences; 7.3 A Construction of Perfect Hash Families; 7.4 Hash Families and Authentication Schemes; Chapter 8 Applications to Low-Discrepancy Sequences This book has two main aims. Firstly, to give a summary of the theoretical work in algebraic curves over finite fields with many rational points. Secondly, to discuss the applications of this theory to areas such as information theory (algebraic coding theory) and computational mathematics (construction of low-discrepancy sequences). The bulk of the material in this book is of very recent origin. The authors have given the first systematic treatment of this material in this book 8.1 Background on (t, m, s)-Nets and (t, s)-Sequences8.2 The Digital Method; 8.3 A Construction Using Rational Places; 8.4 A Construction Using Arbitrary Places; Appendix A: Curves and Their Function Fields; A.1 Transcendence Degree; A.2 Affine Spaces; A.3 Projective Spaces; A.4 Affine Varieties; A.5 Projective Varieties; A.6 Projective Curves; Bibliography; Index Some basic definitions and fundamental properties of algebraic function fields are introduced in this chapter.