An algorithm to compute the $t$-value of a digital net and of its projections

TL;DR

This paper introduces an efficient algorithm for computing the $t$-value of digital nets and their projections, enhancing the assessment of their quality in quasi-Monte Carlo methods.

Contribution

The paper presents a novel, efficient algorithm for calculating $t$-values of digital nets and their projections, improving upon previous methods.

Findings

The new algorithm reduces computational complexity.

Empirical tests show improved accuracy and speed.

Comparison with existing methods demonstrates advantages.

Abstract

Digital nets are among the most successful methods to construct low-discrepancy point sets for quasi-Monte Carlo integration. Their quality is traditionally assessed by a measure called the $t$-value. A refinement computes the $t$-value of the projections over subsets of coordinates and takes a weighted average (or some other function) of these values. It is also of interest to compute the $t$-values of embedded nets obtained by taking subsets of the points. In this paper, we propose an efficient algorithm to compute such measures and we compare our approach with previously proposed methods both empirically and in terms of computational complexity.