Non-Regular Square Bipartite Designs

A square bipartite design is a pair of square 0-1 matrices A and B satisfying the following matrix equation: ATB D J C diag.d/; where J is a matrix filled with all ones, is a positive integer, and diag.d/ is a diagonal matrix with the vector d in the diagonal. We give a characterization of a class of non-regular square bipartite designs; our result generalizes some earlier results of De Bruijn and Erd"os, Lehman, and Gasparyan.