Constructions of codes and low-discrepancy sequences using global function fields

Abstract

The thesis represents a contribution to the theory of global function fields and their applications. We will introduce new constructions of codes and low-discrepancy sequences and produce stronger asymptotic bounds for both. We begin by looking at recent developments in the theory of algebraic-geometry codes such as the use of places of arbitrary degree, distinguished divisors, and local expansions. It will be shown that XNL codes can be used to gain improvements on the Tsfasman-Vladut-Zink bound and, more importantly, a new construction of codes will be introduced which offer improvements on all previous asymptotic coding bounds. We will also show that current constructions of algebraic-geometry codes, (t,m,s)-nets, and (t,s)-sequences all have analogous constructions using differentials. Finally, we show that new developments in the theory of global function fields allow us to gain improved upper bounds on the quality parameter of (t,s)-sequences, which has implications for star discrepancy and numerical integration.