On a problem of Hering concerning orthogonal covers of Kn

A Hering configuration of type k and order n is a factorization of the complete diagraph Kn into n factors each of which consists of an isolated vertex and the edge-disjoint union of directed k-cycles, which has the additional property that for any pair of distinct factors, say Gi, and Gj, there is precisely one pair of vertices, say {ita, b}, such that Gi contains the directed edge (a, b) and Gj contains the directed edge (b, a). Clearly a necessary condition for a Hering configuration is n  1 (mod k). It is shown here that for any fixed k, this condition is asymptotically, and, it is shown to be always sufficient for k = 4.