Linear congruential generators of order K>1

We present a class of random number generators defined by a linear congruential recursion of order k ≥ 1, modulo m, and whose period can attain m^k - 1. The spectral test can be extended to this class of generators, and permits the computation of the distance between the successive t-dimensional hyperplanes in which lie all the t-tuples of successive values. In terms of the spectral test, these generators could be almost as good as the best regular (order 1) linear congruential generators with modulus close to m^k - 1. We discuss the implementation of computer programs to search for maximal period generators of this kind and to apply the (generalized) spectral test. We also present the results of a search to find good vectors of multipliers, for some values of the modulus m, and give an example of a portable implementation.