Designs on vector spaces constructed using quadratic forms

Abstract

Let N be the set of nonnegative integers, let λ, t, v be in N and let K be a subset of N, let V be a v-dimensional vector space over the finite field GF(q), and let W _Kbe the set of subspaces of V whose dimensions belong to K. A t-[v, K, λ; q]-design on V is a mapping ℱ: W _K→ N such that for every t-dimensional subspace, T, of V, we have $\sum _{T \subseteq B}^{B \in WK}$ ℱ (B)=λ. We construct t-[v, {t, t+1}, λ; q-designs on the vector space GF(q ^v) over GF(q) for t≥2, v odd, and q ^t(q−1)²λ equal to the number of nondegenerate quadratic forms in t+1 variables over GF(q). Moreover, the vast majority of blocks of these designs have dimension t+1. We also construct nontrivial 2-[v, k, λ; q]-designs for v odd and 3≤k≤v−3 and 3-[v, 4, q ⁶+q ⁵+q ⁴; q]-designs for v even. The distribution of subspaces in the designs is determined by the distribution of the pairs (Q, a) where Q is a nondegenerate quadratic form in k variables with coefficients in GF(q) and a is a vector with elements in GF(q ^v) such that Q(a)=0.