Spectral test of the MIXMAX random number generators

Abstract

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 224 and 236, 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 optimized range.