Geometric invariants for quasi-symmetric designs.

Let p be an odd prime. We derive new necessary conditons for the existence of 2-(nu, k, lambda) designs where the block intersection sizes s sup 1,s sub 2,...,s sub n satisfy s sub 1=s sub 2=...=s sub n (mod p). The method is to define a non- degenerate scalar product on a 2m-dimensional vector space and to construct an m-dimensional totally singular subspace. This result is a generalization to non-symmetric designs of the Bruck-Ryser-Chowla Theorem.