Separating hash families: A Johnson-type bound and new constructions

TL;DR

This paper advances the theory of separating hash families by establishing a Johnson-type bound, providing new constructions, verifying conjectures on maximal sizes, and connecting these structures to hypergraph Turán problems.

Contribution

It introduces a Johnson-type inequality for separating hash families, constructs infinite classes of perfect hash families, verifies conjectures on their maximal sizes, and links these concepts to hypergraph Turán problems.

Findings

New upper bound for separating hash families surpassing previous bounds.

Construction of an infinite class of perfect hash families based on Hamming graphs.

Verification that p_3(3,q) and p_4(4,q) grow faster than q^{2-o(1)}.

Abstract

Separating hash families are useful combinatorial structures which are generalizations of many well-studied objects in combinatorics, cryptography and coding theory. In this paper, using tools from graph theory and additive number theory, we solve several open problems and conjectures concerning bounds and constructions for separating hash families. Firstly, we discover that the cardinality of a separating hash family satisfies a Johnson-type inequality. As a result, we obtain a new upper bound, which is superior to all previous ones. Secondly, we present a construction for an infinite class of perfect hash families. It is based on the Hamming graphs in coding theory and generalizes many constructions that appeared before. It provides an affirmative answer to both Bazrafshan-Trung's open problem on separating hash families and Alon-Stav's conjecture on parent-identifying codes. Thirdly, let $p_t(N,q)$ denote the maximal cardinality of a $t$-perfect hash family of length $N$ over an alphabet of size $q$. Walker II and Colbourn conjectured that $p_3(3,q)=o(q^2)$. We verify this conjecture by proving $q^{2-o(1)}<p_3(3,q)=o(q^2)$. Our proof can be viewed as an application of Ruzsa-Szemer{\'e}di's (6,3)-theorem. We also prove $q^{2-o(1)}<p_4(4,q)=o(q^2)$. Two new notions in graph theory and additive number theory, namely rainbow cycles and $R$-sum-free sets, are introduced to prove this result. These two bounds support a question of Blackburn, Etzion, Stinson and Zaverucha. Finally, we establish a bridge between perfect hash families and hypergraph Tur{\'a}n problems. This connection has not been noticed before. As a consequence, many new results and problems arise.