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Spectral test of the MIXMAX random number generators. (English) Zbl 1442.65003
Summary: 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 optimized range.
##### MSC:
 65C10 Random number generation in numerical analysis 11K45 Pseudo-random numbers; Monte Carlo methods
MIXMAX; TestU01
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