Incomplete perfect Mendelsohn designs with block size four

AbstractLet v,k,λ and n be positive integers. (x1,x2,…,xk) is defined to be {(xi,xj):i≠j,i,j=1,2,…,k}, in which the ordered pair (xi,xj) is called (j−i)-apart for i<j and (k−j+i)-apart for i>j, and is called a cyclically ordered k-subset of {x1,x2,…,xk}. An incomplete perfect Mendelsohn design, denoted by (v,n,k,λ)-IPMD is a triple (X,Y,B), where X is a v-set (of points), Y is an n-subset of X, and B 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 λ blocks of B and no pair (a,b)∈Y×Y appears in any block of B for any t, where 1⩽t⩽k−1. In this article, we shall show that the necessary conditions v(v−1)−n(n−1)≡0(mod4),v⩾3n+1 for the existence of a (v,n,4,1)-IPMD (n⩾2) are sufficient except for n=2,v−n=5 and possibly excepting n=2,v−n=17,25