# Interpreting “good figures of merit” in xorshift* context

I am trying to implement an xorshift* PRNG (practically an xorshift with a multiplication step in the end) as a long term replacement of Java's Math.Random. I have been reading an article pointed out in an answer to another question, but I am confused with the magical $M_t$ values used for the multiplication.

The article comments on this number in the following way:

The goodness of the multiplier, however, is established by a fi gure of merit which is a normalized best distance between the hyperplanes of families covering tuples of length $t$ given by successive outputs of the generators. The length t is an additional parameter, and $M_{32}$ has the best figures of merit for $t = 32$. Clearly, if an alternative multiplier provides improvements on both t and the associated fi gure of merit, we have a hint that it could be chosen instead of $M_{32}$.

Lacking that possibility, what if we scramble xorshift64's output with a multiplier that has better fi gures of merit for a lower $t$? We thus also ran experiment with the multiplier $M_8 = 1181783497276652981$, which has a better fi gure of merit for $t = 8$ [L'Ecuyer 1999], and the multiplier $M_2 = 8372773778140471301$, which has a better fi gure of merit for $t = 2$ and was kindly provided by Richard Simard.

Seeing as the author has used different multipliers in different xorshift* implementations (different state size) I am not sure how to interpret the role of $t$ in $M_t$. Here's the article by L'Ecuyer that's referenced above.

The paper is written using three multipliers to check experimentally if the $t$ parameter has any influence on the result. For all I could see, there is no such influence. Indeed, it could be even possible that a random invertible element of $\mathbf Z/2^{64}\mathbf Z$ works as well (even if it doesn't have good spectral properties). The choice of a multiplier with good spectral properties comes from a suggestion in "Numerical Recipes".
There is no connection between $t$ and the state-space size. $t$ is an independent parameter. In the paper I try all combinations.