Paper 2005/465

A sequence approach to constructing perfect hash families

S. G. Barwick and W. -A. Jackson

Abstract

A linear $(q^d,q,t)$-perfect hash family of size $s$ in a vector space $V$ of order $q^d$ over a field $F$ of order $q$ consists of a set $\phi_1,\ldots,\phi_s$ of linear functionals from $V$ to $F$ with the following property: for all $t$ subsets $X\subseteq V$ there exists $i\in\{1,\ldots,s\}$ such that $\phi_i$ is injective when restricted to $F$. A linear $(q^d,q,t)$-perfect hash family of minimal size $d(t-1)$ is said to be {\em 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 $(q^2,q,5)$-perfect hash families and $(q^4,q,3)$-perfect hash families. The sequence approach also explains a relationship between linear $(q^3,q,3)$-perfect hash families and linear $(q^2,q,4)$-perfect hash families.