Quasi-affine symmetric designs

Abstract

A symmetric design with parameters v = q 2(q + 2), k = q(q + 1), ? = q, q ? 2, is called a quasi-affine design if its point set can be partitioned into q + 2 subsets P 0, P 1,..., P q , P q+1 such that the induced structure in every point neighborhood is an affine plane of order q (repeated q times). A quasi-affine design with q ? 3 determines its point neighborhoods uniquely and dual of such a design is also a quasi-affine design. These structural properties pave way for definition of a strongly quasi-affine design and it is also shown that associated with every quasi-affine design is a unique strongly quasi-affine design from which the given quasi-affine design is obtained by certain unique cutting and pasting operation. This investigation also enables us to associate a unique 2-regular graph with q + 2 vertices and in turn, a unique colored partition of the integer q + 2. These combinatorial consequences are finally used to obtain an exponential lower bound on the number of non-isomorphic solutions of such symmetric designs improving the earlier lower bound of 2.