Vandermonde Nets

TL;DR

This paper introduces Vandermonde nets, a new class of digital nets over finite fields, providing explicit constructions with optimal quality parameters in certain dimensions and analyzing their discrepancy and existence results.

Contribution

It defines Vandermonde nets, applies existing methods to analyze their quality and discrepancy, and offers explicit constructions in dimensions up to q+1, advancing digital net theory.

Findings

Existence of small quality parameter Vandermonde nets established.

Discrepancy bounds for Vandermonde nets derived.

Explicit constructions available for dimensions s ≤ q+1.

Abstract

The second author recently suggested to identify the generating matrices of a digital $(t,m,s)$-net over the finite field $F_q$ with an $s \times m$ matrix $C$ over $F_{q^m}$. More exactly, the entries of $C$ are determined by interpreting the rows of the generating matrices as elements of $F_{q^m}$. This paper introduces so-called Vandermonde nets, which correspond to Vandermonde-type matrices $C$, and discusses the quality parameter and the discrepancy of such nets. The methods that have been successfully used for the investigation of polynomial lattice point sets and hyperplane nets are applied to this new class of digital nets. In this way, existence results for small quality parameters and good discrepancy bounds are obtained. Furthermore, a first step towards component-by-component constructions is made. A novelty of this new class of nets is that explicit constructions of Vandermonde nets over $F_q$ in dimensions $s \le q+1$ with best possible quality parameter can be given. So far, good explicit constructions of the competing polynomial lattice point sets are known only in dimensions $s \le 2$.