A Technique for Constructing Symmetric Designs

Let M be a set of incidence matrices of symmetric (v,k,λ)-designs and G a group of mappings M→ M. We give a sufficient condition for the matrix W⊗ M, where Mε M and W is a balanced generalized weighing matrix over G, to be the incidence matrix of a larger symmetric design. This condition is then applied to the designs corresponding to McFarland and Spence difference sets, and it results in four families of symmetric (v,k,λ )-designs with the following parameters k and λ (m and d are positive integers, p and q are prime powers): (i) $$k = q^{2m-1} p^d ,\lambda = (q-1)q^{2m-2} p^{d-1} ,q= \frac{p^{d+1} - 1}{p-1}$$ ; (ii) $$k = \frac{(q^{2m-1} p^d - 1)p^d}{(p-1)(p^d + 1)},\lambda = \frac{{(q^{2m - 2} p^{2d} - 1)p^d }}{{(p - 1)(p^d + 1)}},q = p^{d + 1} + p - 1$$ ; (iii) $$k = 3^d q^{2m - 1} ,\lambda = \frac{{3^d (3^d + 1)q^{2m - 2} }} {2},q = \frac{{3^{d + 1} + 1}} {2} $$ ; (iv) $$k = \frac{{3^d (3^d q^{2m - 1} - 1)}} {{2(3^d - 1)}},\lambda = \frac{{3^d (3^{2d} q^{2m - 2} - 1)}} {{2(3^d - 1)}},q = 3^{d + 1} - 2 $$ .