Bounds for digital nets and sequences

1. Introduction. Currently, the most effective constructions of low-discrepancy point sets and sequences, which are of great importance for quasi-Monte Carlo methods in multidimensional numerical integration, are based on the concept of (t, m, s)-nets and (t, s)-sequences. A detailed theory was developed in Niederreiter [9] (see also [10, Chapter 4] for surveys of this theory). So-called digital nets and sequences are of special interest due to the following two reasons. First, until now all construction methods for (t, m, s)-nets and (t, s)-sequences which are relevant for applications in quasi-Monte Carlo methods are digital methods over certain rings. Second, digital (t, m, s)-nets behave extremely well for the numerical integration of functions which are representable by an in some sense rapidly converging mul-tivariate Walsh series. In a series of papers, Larcher and several co-authors established lattice rules for the numerical integration of multivariate Walsh series by digital nets. We refer to [5] for a concise introduction in the field of Larcher's lattice rules.