Symmetric designs and related configurations

AbstractIn this paper we study combinatorial designs that are characterized by a (0, 1)-matrix A of order n ⩾ 3 that satisfies the matrix equation ATA = D + [λiλj] where AT denotes the transpose of A, D denotes the diagonal matrix D = diag[k1 − λ1, k2 − λ2, …, kn − λn] and the scalars ki − λi and λj are positive. We call these configurations multiplicative designs. They are a natural generalization of the classical symmetric block designs and the recently investigated λ-designs. We develop certain basic properties of multiplicative designs. But the complete structure of these interesting configurations is far from determined