Constructingc-ary Perfect Factors

Ac-ary Perfect Factor is a set of uniformly long cycles whose elements are drawn from a set of sizec, in which every possiblev-tuple of elements occurs exactly once. In the binary case, i.e. wherec=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 wheneverc is a prime power. In this paper we conjecture that the same is true for arbitrary values ofc, and exhibit a number of constructions. We also construct a family of related combinatorial objects, which we callPerfect Multi-factors.