Existence of HPMDs with Block Size Five and Index $\lambda \ge 2$

Abstract

In this paper, it is shown that the necessary condition for the existence of a holey perfect Mendelsohn design (HPMD) with block size 5, type $h^n$ and index $\lambda$, namely, $n\ge 5$ and $\lambda n(n-1)h^2\equiv 0(\mathrm{mod}5)$, is also sufficient for $\lambda \ge 2$. The result guarantees the analogous existence result for group divisible designs (GDDs) of type $h^n$ having block size 5 and index $4\lambda$.