Rational Points on Curves over Finite Fields Theory and Applications [Paperback]

Discussion of theory and applications of algebraic curves over finite fields with many rational points.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.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.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.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.'& the book under review develops many techniques that are not covered in the existing texts. I highly recommend it.' Steven D. Galbraith, Royal Holl,