Constructing Sobol Sequences with Better Two-Dimensional Projections

Abstract

Direction numbers for generating Sobol$'$ sequences that satisfy the so-called Property A in up to 1111 dimensions have previously been given in Joe and Kuo [ACM Trans. Math. Software, 29 (2003), pp. 49–57]. However, these Sobol$'$ sequences may have poor two-dimensional projections. Here we provide a new set of direction numbers alleviating this problem. These are obtained by treating Sobol$'$ sequences in d dimensions as $(t,d)$-sequences and then optimizing the t-values of the two-dimensional projections. Our target dimension is 21201.