The following xorshift+ generator, instead, has 128 bits of state, a maximal period of 2^128 − 1 and passes BigCrush: [snip code] This generator is one of the fastest generator passing BigCrush; however, it is only 1-dimensionally equidistributed.
Earlier in the article there’s the following:
[snip code] Both generators, as all xorshift* generators of maximal period, emit a sequence of 64-bit values that is equidistributed in the maximum possible dimension.
What does it mean for a sequence to be equidistributed in one dimension vs. multiple dimensions vs. not at all?
Asked By : Claudiu
Answered By : Claudiu
A xorshift* generator with an n-bit state space is n/64-dimensionally equidistributed: every n/64-tuple of consecutive 64-bit values appears exactly once in the output, except for the zero tuple (and this is the best possible for 64-bit values). A xorshift+ generator is however only (n/64 − 1)-dimensionally equidistributed: every (n/64 − 1)-tuple of consecutive 64-bit values appears exactly 264 times in the output, except for a missing zero tuple.
In the case of a xorshift+ which uses 128 bits of state, n/64 – 1 = 128/64 – 1 = 2 – 1 = 1, so the Wikipedia article was correct in stating that that particular generator is only 1-dimensionally equidistributed.
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