Some intriguing upper bounds for separating hash families

TL;DR

This paper introduces two new methodologies combining graph theory, coding theory, and probabilistic techniques to derive improved upper bounds for the size of separating hash families, advancing understanding in combinatorial design theory.

Contribution

The paper presents two novel approaches for establishing upper bounds on separating hash families, integrating graph removal lemmas, Johnson-type inequalities, and probabilistic methods.

Findings

Derived new upper bounds for separating hash families.

Significantly improved previous bounds for specific parameters.

Provided methodologies applicable to related combinatorial problems.

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.