On the existence of hyperplane sequences, with quality parameter and discrepancy bounds

TL;DR

This paper introduces hyperplane sequences, a new class of digital sequences generalizing existing nets, and analyzes their distribution properties and discrepancy bounds for quasi-Monte Carlo methods.

Contribution

It presents the construction of hyperplane sequences based on duality theory, extending the class of digital sequences with proven distribution and discrepancy properties.

Findings

Hyperplane sequences have good equidistribution properties.

They achieve low discrepancy bounds.

The construction generalizes known digital nets.

Abstract

It is well-known that digital $(t,m,s)$-nets and $(\Tfett,s)$-sequences over a finite field have excellent properties when they are used as underlying nodes in quasi-Monte Carlo integration rules. One very general sub-class of digital nets are hyperplane nets which can be viewed as a generalization of cyclic nets and of polynomial lattice point sets. In this paper we introduce infinite versions of hyperplane nets and call these sequences hyperplane sequences. Our construction is based on the recent duality theory for digital sequences according to Dick and Niederreiter. We then analyze the equidistribution properties of hyperplane sequences in terms of the quality function $\Tfett$ and the star discrepancy.