In the study of algorithms for pseudorandom numbers, an important factor is the usability of a sequence of pseudorandom numbers for particular purposes. Their use in a periodic fashion is frequent, and so the statistical independence of k-tuples of successive terms for particular ks is an important factor in choosing such an algorithm. (The periodicity of the sequence implies the existence of some ks, albeit hopefully rather large, for which independence does not hold.) The paper uses Weyl’s necessary and sufficient condition for k:-distributivity, defined in terms of Walsh functions. It applies a discrete version of the formula to GFSR sequences, that is, those derived by the successive multiplication of the vector of the binary digits of the preceding number by a non-singular matrix over GF(2). This leads to an explicit criterion for a regular k-distribution that is easy to compute.

The criterion is applied to a number of specific generators, and there is some further discussion of other uses of the test. The paper is easy to read, and the result will interest those intending to use pseudorandom number generators to produce numbers to be used in regular fashions in their work.