Existence of HPMDs with Block Size Five and Index \(\lambda \geq 2\)

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, \(n \geq 5\) and \(\lambda n(n-1)h^2 \equiv 0 \pmod{5}\), is also sufficient for \(\lambda \geq 2\). The result guarantees the analogous existence result for group divisible designs (GDDs) of type \(h^n\) having block size 5 and index \(4\lambda\).