Perfect Hash Families with Few Functions

An (s; n; q; t)-perfect hash family is a set of functions 1 ; 2 ; : : :; 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 i. The paper shows that the maximum value n s;t (q) that n can take for xed 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 coeecient of this linear leading term; this coeecient is explicitly calculated in some cases. As part of this process, new classes of good perfect hash families are constructed.