Geodesic
Conditions for singular incidence matrices
Journal of Algebraic Combinatorics, Tome 21 (2005) no. 2, pp. 179-183.

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Summary: 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.
Keywords: keywords incidence matrix, bruck-Ryser-chowla theorem, generalised polygon, divisible design, distance-regular graph 
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     title = {Conditions for singular incidence matrices},
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Haemers, Willem H. Conditions for singular incidence matrices. Journal of Algebraic Combinatorics, Tome 21 (2005) no. 2, pp. 179-183. http://geodesic.mathdoc.fr/item/JAC_2005__21_2_a1/