Incomplete perfect mendelsohn designs with block size 4 and one hole of size 7

Let v,k , and n be positive integers. An incomplete perfect Mendelsohn design, denoted by k ‐IPMD (v,n) , is a triple ( X, Y , ) where X is a v ‐set (of points), Y is an n ‐subset of X , and is a collection of cyclically ordered k ‐subsets of X (called blocks ) such that every ordered pair ( a, b ) ∈ ( X × X )∖( Y × Y ) appears t ‐apart in exactly one block of and no ordered pair ( a,b ) ∈ Y × Y appears in any block of for any t , where 1 ≤ t ≤ k − 1. In this article, we obtain conclusive results regarding the existence of 4‐IPMD( v ,7) where the necessary conditions are v = 2 or 3(mod 4) and v ≥ 22. We also provide an application to the problem relating to coverings of PMDs with block size 4. © 1993 John Wiley & Sons, Inc.