Bad Lattice Structures for Vectors of Nonsuccessive Values Produced by Some Linear Recurrences

Usually, the t-dimensional spectral test for linear congruential generators examines the lattice structure of all the points formed by taking t successive values in the sequence. In this paper, we consider the case where the t values taken are not successive, but separated by lags that are chosen a priori. For certain classes of linear congruential and multiple recursive generators, and for certain choices of the lags, we give lower bounds on the distance between hyperplanes. In some cases, those lower bounds are quite large, even in dimensions as small as t = 3. We give illustrations with speciic classes of generators that have been proposed in the literature, and discuss the possible implications.