Perfect Mendelsohn designs with block size six

AbstractLet v, k and λ be positive integers. A perfect Mendelsohn design with parameters v, k and λ, denoted by (v, k, λ)-PMD, is a decomposition of the complete directed multigraph λkv* on v vertices into k-circuits such that for any r, 1 ⩽ r ⩽ k − 1, and for any two distinct vertices x and y there are exactly λ circuits along which the (directed) distance from x to y is r. It is known that a (6, 6, 1)-PMD does not exist. In this paper we show that a (v, 6, 1)-PMD exists for any v > 6, where v ≠ 0 or 1 (mod 6), with at most 150 possible exceptions of which 2604 is the largest