Combinatorial designs with two singular values—I: uniform multiplicative designs

In this and a sequel paper (Combinatorial designs with two singular values. II. Partial geometric designs, preprint) we study combinatorial designs whose incidence matrix has two distinct singular values. These generalize 2-(v,k,λ) designs, and include partial geometric designs and uniform multiplicative designs. Here we study the latter, which are precisely the nonsingular designs. We classify all such designs with smallest singular value at most 2, generalize the Bruck–Ryser–Chowla conditions, and enumerate, partly by computer, all uniform multiplicative designs on at most 30 points.