Thin Lehman matrices arising from finite groups

Two n×n (0,1) matrices X, Y are called thin Lehman matrices if they are solutions of the matrix equation XYT=Jn+In, where Jn is the n×n matrix of all 1s and In is the identity matrix. These matrices are important in the set covering problem, but few examples are known. In this paper, we will introduce the notion of 1-overlapped factorizations of finite groups which constructs a new class of thin Lehman matrices. Moreover, we will study some structural properties of 1-overlapped factorizations.