An Equivalence between (T, M, S)-Nets and Strongly Orthogonal Hypercubes

It is well known that (t, m, s)-nets are useful in numerical analysis. While many of the best constructions of such nets arise from number theoretic or algebraic constructions, we will show in this paper that the existence of a (t, t+k, s)-net in basebis equivalent to the existence of a set of s strongly orthogonal hypercubes of dimensiont+k, orderband strengthk. Fork>2 such generalized orthogonal hypercubes provide new combinatorial structures that may be of interest in various other combinatorial settings.