Existence of Higher Order Convergent Quasi-Monte Carlo Rules via Walsh Figure of Merit

Abstract

The Walsh figure of merit WAFOM(P) is a quality measure of point sets $P \subset [0,1{)}^{S}$ in the S-dimensional unit cube for quasi-Monte Carlo integration constructed by a digital net method with n-bit precision over the two element field. We prove that there are explicit constants E, C, D such that for any $d \geq 9S$ and n, there is a point set P of size N: = 2^d with $\mathrm{WAFOM}(P) \leq E \cdot {2}^{-{\mathit{Cd}}^{2}/S+\mathit{Dd} } = E \cdot {N}^{-C(\log _{2}N)/S+D}$, by bounding WAFOM(P) by the minimum Dick-weight of ${P}^{\perp }$, and by proving the existence of point sets with large minimum Dick-weight by a probabilistic argument.