Lehman matrices

A pair of square 0 , 1 matrices A , B such that A B T = E + k I (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 Lehmanʹs 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 = k 2 + 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.