An arithmetic version of the crosscorrelation of two sequences is defined, generalizing Mandelbaum’s arithmetic autocorrelations. Large families of sequences are constructed with ideal (vanishing) arithmetic cross-correlations. These sequences are decimations of the 2-adic expansions of rational numbers p/q such that 2 is a primitive root modulo q.

In this paper, we introduce a vectorial conception of dFCSRs to build these registers over any finite field. We describe the structure of d-vectorial FCSRs and we develop an analysis to obtain basic properties like periodicity and the existence of maximal length sequences. To illustrate these vectorial d-FCSRs, we present simple examples and we compare with those of Goresky, Klapper and Xu.

In this paper we study the distribution properties of d-FCSR sequences. These sequences have efficient generators and have several good statistical properties. We show that for d = 2 the number of occurrences of an fixed size subsequence differs from the average number of occurrences by at most a small constant times the square root of the average.

The k-set agreement problem is a generalization of the consensus problem. Namely, assuming that each process proposes a value, every non-faulty process should decide one of the proposed values, and no more than k different values should be decided. This is a hard problem in the sense that we cannot solve it in an asynchronous system, as soon as k or more processes may crash. One way to sidestep...