Some intriguing upper bounds for separating hash families

Abstract

An $N\times n$ matrix on $q$ symbols is called $\{w_1,\ldots,w_t\}$-separating if for arbitrary $t$ pairwise disjoint column sets $C_1,\ldots,C_t$ with $|C_i|=w_i$ for $1\le i\le t$, there exists a row $f$ such that $f(C_1),\ldots,f(C_t)$ are also pairwise disjoint, where $f(C_i)$ denotes the collection of components of $C_i$ restricted to row $f$. Given integers $N,q$ and $w_1,\ldots,w_t$, denote by $C(N,q,\{w_1,\ldots,w_t\})$ the maximal $n$ such that a corresponding matrix does exist. The determination of $C(N,q,\{w_1,\ldots,w_t\})$ has received remarkable attentions during the recent years. The main purpose of this paper is to introduce two novel methodologies to attack the upper bound of $C(N,q,\{w_1,\ldots,w_t\})$. 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 $C(N,q,\{w_1,\ldots,w_t\})$, which significantly improve the previously known results.