A Coding Theoretic Approach to Building Nets with Well-Equidistributed Projections,

coauthor Pierre L'Ecuyer

Monte Carlo and Quasi-Monte Carlo Methods 2006, S. Heinrich, A. Keller, and H. Niederreiter Eds., Springer-Verlag, 2008, 313-325.



Starting from coding-theoretic constructions, we build digital nets with good figures of merit, where the figure of merit takes into account the equidistribution of a preselected set of low-dimensional projections. This type of figure of merit turns out to be a better predictor than the t-value for the variance of randomized quasi-Monte Carlo (RQMC) estimators based on nets, for certain classes of integrals. Our construction method determines the most significant digits of the points by exploiting the equivalence between the desired equidistribution properties used in our criterion and the property of a related point set to be an orthogonal array, and using existing orthogonal array constructions. The least significant digits are then adjusted to improve the figure of merit. Known results on orthogonal arrays provide bounds on the best possible figure of merit that can be achieved. We present a concrete construction that belongs to the class of cyclic digital nets and we provide numerical illustrations of how it can reduce the variance of a RQMC estimator, compared with more standard constructions.

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