Linear recurrences with carry as uniform random number generators

We study the multiply-with-carry family of generators proposed by G. Marsaglia (1994) as a generalisation of the previous add-with-carry and subtract-with-borrow families of G. Marsaglia and A. Zaman (1991). We define for them a general (infinite) state space and focus our attention on the (finite) subset of recurrent states. This subset will, in turn, split into possibly several subgenerators. We discuss the uniformity of the d-dimensional distribution of the output of these subgenerators over their full period. In order to improve this uniformity for higher dimensions, we propose a method for finding good parameters in terms of the spectral test.