Paper 2003/017

Perfect Hash Families with Few Functions

Simon R. Blackburn

Abstract

An {\em $(s;n,q,t)$-perfect hash family} is a set of functions $\phi_1,\phi_2,\ldots ,\phi_s$ from a set $V$ of cardinality $n$ to a set $F$ of cardinality $q$ with the property that every $t$-subset of $V$ is injectively mapped into $F$ by at least one of the functions $\phi_i$. The paper shows that the maximum value $n_{s,t}(q)$ that $n$ can take for fixed $s$ and $t$ has a leading term that is linear in $q$ if and only if $t>s$. Moreover, for any $s$ and $t$ such that $t>s$, the paper shows how to calculate the coefficient of this linear leading term; this coefficient is explicitly calculated in some cases. As part of this process, new classes of good perfect hash families are constructed.

Note: This paper is about to undergo a major rewrite. In particular, a reference to a paper of Fachini and Nilli (Recursive bounds for perfect hashing, Discrete Applied Maths 2001) that appeared after the paper has written will be added. Fachini and Nilli's improvement of a bound of Dyachkov can be used as the `if' part of Theorem 1 of my paper (and their argument is essentially the same as the one I give).