Constructing c-ary Perfect Factors

A c-ary Perfect Factor is a set of uniformly long cycles whose elements are drawn from a set of size c, in which every possible v-tuple of elements occurs exactly once. In the binary case, i.e. where c = 2, these perfect factors have previously been studied by Etzion, [2], who showed that the obvious necessary conditions for their existence are in fact sufficient. This result has recently been extended by Paterson, [4], who has shown that the necessary existence conditions are sufficient whenever c is a prime power. In this paper we conjecture that the same is true for arbitrary values of c, and exhibit a number of constructions. We also construct a family of related combinatorial objects, which we call Perfect Multi-factors.