Rational Points on Curves over Finite Fields Theory and Applications

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.

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.

This book gives a summary of the theory and discusses the applications of algebraic curves over finite fields with many rational points. The bulk of the material in this book is of very recent origin and has not been systematically treated in any other works.