Algebraic function fields over finite fields with many rational places

Abstract

Algebraic function fields (or equivalently, algebraic curves) provide a useful tool for coding theory (for instance, algebraic-geometric codes and trace codes), but also for other branches of information theory. In these applications, the number of rational places of a function field plays a crucial role. One is particularly interested in function fields having a large number of rational places. After a short introduction into the mathematical theory of algebraic functions, the paper gives a survey of old and new results on the number of rational places of function fields.