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.

‘… the book under review develops many techniques that are not covered in the existing texts. I highly recommend it.’

Steven D. Galbraith - Royal Holloway, University of London

'Because of the carefully selected contents and lucid style, the book can be warmly recommended to mathematicians interested in the above-mentioned topics or in algebraic curves over finite fields with many rational points.'

Source: EMS

'The book is very clearly written. It is warmly recommended to anyone who is interested in nice mathematical theories and/or in the recent applications.'

Source: Acta. Sci. Math.