A Large Set of Designs On Vector Spaces Original Research Article

A t-design on a vector space V over the finite field GF(q) is a multiset image of subspaces of V over GF(q) such that for every t-dimensional supspace T of V, exactly λ many subspaces in image, counting according to multiplicity, contain T. Let our vector space be GF(qv) over GF(q). We use the nondegenerate quadratic forms to partition the subspaces of GF(qv) over GF(q) according to the action of the quadratic forms on the bases of the subspaces. This partitioning yields a family of t-designs on GF(qv) with block dimensions t and t + 1. The union of these designs is the union of an n-fold multiple of all (t + 1)-dimensional subspaces and an n/(q − 1)-fold multiple of all t-dimensional subspaces, where n is the number of nondegenerate quadratic forms in t + 1 variables over GF(q). We investigate the structure of the 1- and 2-designs in detail, showing that most of the 1-designs are near-spreads.