A combinatorial characterization of (t,m,s)-nets in baseb

Abstract

We consider (t,m,s)-nets in base b, which were introduced by Niederreiter in 1987. These nets are highly uniform point distributions in s-dimensional unit cubes and have applications in the theory of numerical integration and pseudorandom number generation. A central question in their study is the determination of the parameter values for which these nets exist. Niederreiter has given several methods for their construction, all of which are based on a general construction principle from his 1987 paper. We define a new family of combinatorial objects, the so-called “generalized orthogonal arrays,” and then discuss a combinatorial characterization of (t.m.s)-nets in base b in terms of these generalized orthogonal arrays. Using this characterization, we describe a new method for constructing (t.m.s)-nets in base b that is not based on the aforementioned construction principle. This method gives rise to some very general conditions on the parameters (involving a link with the theory of orthogonal arrays) that are sufficient to ensure the existence of a (t.m.s)-net in base b. In this way we construct many nets that are new. © 1996 John Wiley & Sons, Inc.