Pre-Scrambled Digital Nets for Randomized Quasi-Monte Carlo

When using digital nets for randomized quasi-Monte Carlo, the generating matrices are usually randomly scrambled before applying a random digital shift. This scrambling is to remove the excessive structure that the original matrices may have. In this paper, we explore the idea of pre-scrambling the generating matrices to “optimize” them in some way so that applying the random digital shift alone becomes sufficient. For the optimization criterion, we experiment with a class of recently-proposed figures of merit based on truncated versions of error and variance bounds obtained by bounding the coefficients in the Walsh expansion of smooth integrands. We summarize our numerical experiments and some difficulties encountered.