A Product Theorem for Intersection Families

We will consider a certain kind of intersection family: a family of vectors over a field of finite characteristic q, such that the generalized inner products of every q or fewer vectors take on prescribed values. For certain values of the parameters, these values distinguish the generalized inner product of any q distinct vectors, from that of any q vectors in which some vector is repeated. l represent such an intersection family by its incidence matrix, and show that the matrix product of two such matrices is itself an intersection family of the type under consideration. If the factor families distinguish between generalized inner products of q vectors with or without repetition, then so does the product family.