On the distribution of fc-dimensional vectors for simple and combined tausworthe sequences

Abstract

The lattice structure of conventional linear congruential random number 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) evolves according to a linear recursion and can be mapped to a number between 0 and 1, producing what we call a LS2 sequence. In particular, the 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 distributed in the unit hypercube. More specifically, for any partition of the fc-dimensional hypercube into 2klidentical subcubes, we can quickly compute 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. © 1993 American Mathematical Society.