A Lower Bound on WAFOM

TL;DR

This paper establishes an optimal lower bound on WAFOM, a parameter used to estimate integration error in quasi-Monte Carlo methods, confirming the existence of point sets that attain this bound.

Contribution

It provides the first proven lower bound on WAFOM, demonstrating its optimality and confirming the existence of point sets that achieve this bound.

Findings

Established a lower bound on WAFOM.

Proved the bound is optimal by referencing existing constructions.

Confirmed the existence of point sets attaining the bound.

Abstract

We give a lower bound on Walsh figure of merit (WAFOM), which is a parameter to estimate the integration error for quasi-Monte Carlo (QMC) integration by a point set called a digital net. This lower bound is optimal because the existence of point sets attaining the order was proved in [K. Suzuki, An explicit construction of point sets with large minimum Dick weight, Journal of Complexity 30, (2014), 347-354].