What does it mean for a random number generator's sequence to be only 1-dimensionally equidistributed?

Whilst reading up on Xorshift I came across the following (emphases added):

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?

This website provides the answer:


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 $2^{64}$ 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.

StackExchange Computer Science Q#24037, answer score: 5