Conditions for Singular Incidence Matrices

Suppose one looks for a square integral matrix N, for which NN_T has a prescribed form. Then the Hasse-Minkowski invariants and the determinant of NN_T lead to necessary conditions for existence. The Bruck-Ryser-Chowla theorem gives a famous example of such conditions in case N is the incidence matrix of a square block design. This approach fails when N is singular. In this paper it is shown that in some cases conditions can still be obtained if the kernels of N and N_T are known, or known to be rationally equivalent. This leads for example to non-existence conditions for self-dual generalised polygons, semi-regular square divisible designs and distance-regular graphs.