Quasi-symmetric designs and the Smith Normal Form

We obtain necessary conditions for the existence of a 2 – (, k, ) design, for which the block intersection sizes s₁, s₂, ..., s_nsatisfy s₁ s₂ ... s_n s (mod p^e),where p is a prime and the exponent e is odd. These conditions are obtained from restriction on the Smith Normal Form of the incidence matrix of the design. We also obtain restrictions on the action of the automorphism group of a 2 – (, k, ) design on points and on blocks.