Some intriguing upper bounds for separating hash families

Abstract

An N - n matrix on q symbols is called {w1,...,wt}-separating if for arbitrary t pairwise disjoint column sets C1,...,Ct with |Ci| = wi for 1 ≤ i ≤ t, there exists a row f such that f(C1),..., f(Ct) are also pairwise disjoint, where f(Ci) denotes the collection of components of Ci restricted to row f. Given integers N, q and w1,...,wt, denote by C(N, q, {w1,...,wt}) the maximal n such that a corresponding matrix does exist. The determination of C(N, q, {w1,...,wt}) has received remarkable attention during the recent years. The main purpose of this paper is to introduce two novel methodologies to attack the upper bound of C(N, q, {w1,...,wt}). The first one is a combination of the famous graph removal lemma in extremal graph theory and a Johnson-type recursive inequality in coding theory, and the second one is the probabilistic method. As a consequence, we obtain several intriguing upper bounds for some parameters of C(N, q, {w1,...,wt}), which significantly improve the previously known results.