Explicit constructions of Vandermonde sequences using global function fields

TL;DR

This paper presents explicit methods for constructing Vandermonde sequences over global function fields, extending their size and dimension, which are useful for digital nets in quasi-Monte Carlo methods.

Contribution

It introduces an algorithm for infinite-dimensional Vandermonde sequences over rational function fields and generalizes the approach to global function fields of positive genus.

Findings

Constructed infinite matrices over rational function fields.

Extended Vandermonde sequence constructions to higher dimensions.

Provided explicit algorithms for sequence generation.

Abstract

The authors recently introduced so-called Vandermonde nets. These digital nets share properties with the well-known polynomial lattices. For example, both can be constructed via component-by-component search algorithms. A striking characteristic of the Vandermonde nets is that for fixed $m$ an explicit construction of $m \times m$ generating matrices over the finite field $F_q$ is known for dimensions $s \le q+1$. This paper extends this explicit construction in two directions. We give a maximal extension in terms of $m$ by introducing a construction algorithm for $\infty \times \infty$ generating matrices for digital sequences over $F_q$, which works in the rational function field over $F_q$. Furthermore, we generalize this method to global function fields of positive genus, which leads to extensions in the dimension $s$.