On applications of algebraic function fields to codes

Official URL: http://risc01.sabanciuniv.edu/record=b1124923 (Table of Contents)

The relation between algebraic function fields over finite fields and coding theory started with Goppa's important code construction, which is nowadays called geometric Goppa codes. He used Riemann-Roch spaces of divisors and degree one (rational) places of a function field to write codes with good parameters. Since Goppa's work, interaction between function fields and codes has been investigated extensively and further applications in coding theory have been found. The aim of this thesis is to describe two of these applications. The first is Goppas idea and its generalization by Xing-Niederreiter-Lam and Heydtmann using higher degree places of the function field. The second application is the use of number of rational places of a function field to estimate the minimum distance of cyclic codes. We give two examples of cyclic codes; binary Hamming and BCH codes.