**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.

**Category / Keywords: **applications / perfect hash families

**Date: **received 3 Jan 2006

**Contact author: **sue barwick at adelaide edu au

**Available format(s): **PDF | BibTeX Citation

**Version: **20060104:072540 (All versions of this report)

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