On the distribution of 𝑘-dimensional vectors for simple and combined Tausworthe sequences

Couture, Raymond; L’Ecuyer, Pierre; Tezuka, Shu

doi:10.1090/S0025-5718-1993-1176708-4

On the distribution of $k$-dimensional vectors for simple and combined Tausworthe sequences
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by Raymond Couture, Pierre L’Ecuyer and Shu Tezuka PDF

Math. Comp. 60 (1993), 749-761 Request permission

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 ${\mathbb {F}_2}$. 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 k-dimensional hypercube into ${2^{kl}}$ identical 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.

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