Halton-type sequences from global function fields

TL;DR

This paper introduces a novel construction of $(t,s)$-sequences using global function fields, providing an analog of Halton sequences that improves discrepancy bounds without relying on digital methods.

Contribution

It presents the first general, non-digital construction of $(t,s)$-sequences from global function fields, enhancing discrepancy bounds within the $(u,e,s)$-sequence framework.

Findings

First non-digital $(t,s)$-sequence construction from global function fields.

Improved discrepancy bounds for the constructed sequences.

Framework integration with $(u,e,s)$-sequences by Tezuka.

Abstract

For any prime power $q$ and any dimension $s$, a new construction of $(t,s)$-sequences in base $q$ using global function fields is presented. The construction yields an analog of Halton sequences for global function fields. It is the first general construction of $(t,s)$-sequences that is not based on the digital method. The construction can also be put into the framework of the theory of $(u,e,s)$-sequences that was recently introduced by Tezuka and leads in this way to better discrepancy bounds for the constructed sequences.