Spectral Test of the MIXMAX Random Number Generators

An important statistical test on the pseudo-random number generators is called the spectral test. The test is aimed at answering the question of distribution of the generated pseudo-random vectors in dimensions $d$ that are larger than the genuine dimension of a generator $N$. In particular, the default MIXMAX generators have various dimensions: $N=8,17,240$ and higher. Therefore the spectral test is important to perform in dimensions $d > 8$ for $N=8$ generator, $d> 17$ for $N=17$ and $d> 240$ for $N=240$ generator. These tests have been performed by L'Ecuyer and collaborators. When $d > N$ the vectors of the generated numbers fall into the parallel hyperplanes and the distances between them can be larger than the genuine "resolution" of the MIXMAX generators, which is $l=2^{-61}$. The aim of this article is to further study the spectral properties of the MIXMAX generators, to investigate the dependence of the spectral properties of the MIXMAX generators as a function of their internal parameters and in particular their dependence on the parameter $m$. We found that the best spectral properties are realized when $m$ is between $2^{24}$ and $2^{36}$, a range which is inclusive of the value of the $N=17$ generator. We also provide the alternative parameters for the generators, $N=8$ and $N=240$ with $m$ in this optimised range.