A sequence approach to 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 set φ1, . . . , φs of linear functionals 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 extend the theory for linear perfect hash families based on sequences developed by Blackburn and Wild. We develop techniques which we use to construct new optimal linear (q2, q, 5)-perfect hash families and (q4, q, 3)perfect hash families. The sequence approach also explains a relationship between linear (q3, q, 3)-perfect hash families and linear (q2, q, 4)-perfect hash families.