Some intriguing upper bounds for separating hash families

Abstract

An $N\times n$ matrix on $q$ symbols is called $\{w_1,\dots ,w_t\}$-separating if for arbitrary $t$ pairwise disjoint column sets $C_1,\dots ,C_t$ with $|C_i|=w_i$ for $1\le i\le t$, there exists a row $f$ such that $f(C_1),\dots ,f(C_t)$ 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 $w_1,\dots ,w_t$, denote by $C(N,q,\{w_1,\dots ,w_t\})$ the maximal $n$ such that a corresponding matrix does exist. The determination of $C(N,q,\{w_1,\dots ,w_t\})$ 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 $C(N,q,\{w_1,\dots ,w_t\})$. 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 $C(N,q,\{w_1,\dots ,w_t\})$, which significantly improve the previously known results.

18 pages, finial version, accepted by SCIENCE CHINA Mathematics

References in corpus (3)

• New Bounds For Frameproof Codes
• Sparse hypergraphs: new bounds and constructions
• Constructions and bounds for separating hash families

Cited by in corpus (2)

• Distributing hash families with few rows
• Constructions and bounds for separating hash families