Paper 2006/002

Geometric constructions of optimal linear 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 sequence $\phi_1,\ldots,\phi_s$ of linear functions 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 optimal. In this paper we use projective geometry techniques to completely determine the values of $q$ for which optimal linear $(q^3,q,3)$-perfect hash families exist and give constructions in these cases. We also give constructions of optimal linear $(q^2,q,5)$-perfect hash families.