A LARGE SET OF DESIGNS ON VECTOR-SPACES

[摘要] A t-design on a vector space V over the finite field GF(q) is a multiset B of subspaces of V over GF(q) such that for every t-dimensional supspace T of V, exactly lambda many subspaces in B, counting according to multiplicity, contain T Let our vector space be GR(q(v)) over GF(q). We use the nondegenerate quadratic forms to partition the subspaces of GF(q(v)) 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(q(v)) 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. (C) 1994 Academic Press, Inc.

[发布日期] 1994-06-01