Lehman matrices

A pair of square 0,1 matrices A,B such that ABT=E+kI (where E is the n×n matrix of all 1s and k is a positive integer) are called Lehman matrices. These matrices figure prominently in Lehman's seminal theorem on minimally nonideal matrices. There are two choices of k for which this matrix equation is known to have infinite families of solutions. When n=k2+k+1 and A=B, we get point-line incidence matrices of finite projective planes, which have been widely studied in the literature. The other case occurs when k=1 and n is arbitrary, but very little is known in this case. This paper studies this class of Lehman matrices and classifies them according to their similarity to circulant matrices.