Global function fields with many rational places

Abstract

This thesis continues the extensive research carried out by Niederreiter and Xing in a series of papers on the search for global function fields with sufficiently many rational places relative to their genera. Such function fields, or equivalently, smooth, absolutely irreducible projective curves over finite fields have direct applications to important areas such as in coding theory and cryptography. Our approach primarily employs the theory of cyclotomic function fields or more generally, Drinfeld modules of rank 1. In particular, by exploiting the connection between error-correcting codes and global function fields, together with the aid of Mathematical software packages, several new global function fields with more rational places than existing fields with the same genera are obtained. Further, we investigate the asymptotic bounds of $A(q)$ and establish improved lower bounds on A(q) for small prime values q including q = 2, 5, 7 and 11.