Geometric constructions of optimal linear perfect hash families

A linear (qd,q,t)-perfect hash family of size s in a vector space V of order qd over a field F of order q consists of a sequence ϕ1,…,ϕs of linear functions from V to F with the following property: for all t subsets X⊆V there exists i∈{1,…,s} such that ϕi is injective when restricted to F. A linear (qd,q,t)-perfect hash family of minimal size d(t−1) is said to be optimal. In this paper we use projective geometry techniques to completely determine the values of q for which optimal linear (q3,q,3)-perfect hash families exist and give constructions in these cases. We also give constructions of optimal linear (q2,q,5)-perfect hash families.