Quasi-Symmetric Designs and the Smith Normal Form.

We obtain necessary conditions for the existence of a 2-(v,k,lambda) design, for which the block intersection sizes s sub 1, s sub 2,...., s sub n satisfy s sub 1 is congruent to s sub 2 is congruent to ... is congruent to s sub n is congruent to s (mod p sup e), where p is a prime and the exponent e is odd. These conditions are obtained from restrictions on the Smith Normal Form of the incidence matrix of the design.