The existence of bimatching designs

A collection of k-matchings of the bipartite graph Kn,n with the property that every pair of independent edges lies in exactly A of the k-matchings is called a BIMATCH(n, k, A)-design. In this paper we give a new construction of bimatching designs, and show that the necessary conditions for the existence of a BIMATCH(n, k, A)-design are also sufficient whenever k = 3 and 4.