ON THE DISTRIBUTION OF K-DIMENSIONAL VECTORS FOR SIMPLE AND COMBINED TAUSWORTHE SEQUENCES

The lattice structure of conventional linear congruential randon, numb er generators (LCGs), over integers, is well known. In this paper, we study LCGs in the field of formal Laurent series, with coefficients in the Galois field F2. The state of the generator (a Laurent series) ev olves according to a linear recursion and can be mapped to a number be tween 0 and 1, producing what we call a LS2 sequence. In particular, t he sequences produced by simple or combined Tausworthe generators are special cases of LS2 sequences. By analyzing the lattice structure of the LCG, we obtain a precise description of how all the k-dimensional vectors formed by successive values in the LS2 sequence are distribute d in the unit hypercube. More specifically, for any partition of the k -dimensional hypercube into 2kl identical subcubes, we can quickly com pute a table giving the exact number of subcubes that contain exactly n points, for each integer n . We give numerical examples and discuss the practical implications of our results.