Vandermonde nets

Roswitha Hofer ¹   ; Harald Niederreiter ²

¹ Institute of Financial Mathematics Johannes Kepler University Linz Altenbergerstr. 69 A-4040 Linz, Austria
² Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenbergerstr. 69 A-4040 Linz, Austria and Department of Mathematics University of Salzburg Hellbrunnerstr. 34 A-5020 Salzburg, Austria

Acta Arithmetica, Tome 163 (2014) no. 2, pp. 145-160

Résumé

The second-named author recently suggested identifying the generating matrices of a digital $(t,m,s)$-net over the finite field ${\mathbb F}_q$ with an $s \times m$ matrix $C$ over ${\mathbb F}_{q^m}$. More exactly, the entries of $C$ are determined by interpreting the rows of the generating matrices as elements of ${\mathbb F}_{q^m}$. 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 ${\mathbb F}_q$ in dimensions $s\leq q+1$ 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 $s\leq 2$.

Détail

DOI : 10.4064/aa163-2-5

Keywords: second named author recently suggested identifying generating matrices digital net finite field mathbb times matrix nbsp mathbb exactly entries determined interpreting rows generating matrices elements mathbb paper introduces so called vandermonde nets which correspond vandermonde type matrices discusses quality parameter discrepancy nets methods have successfully investigation polynomial lattice point sets hyperplane nets applied class digital nets existence results small quality parameters discrepancy bounds obtained furthermore first step towards component by component constructions made novelty class nets explicit constructions vandermonde nets mathbb dimensions leq best possible quality parameter given far explicit constructions competing polynomial lattice point sets known only dimensions leq