Improved Bounds for $(b,k)$-hashing

TL;DR

This paper improves bounds on the size of $(b,k)$-hash families, which are sets with specific combinatorial properties, extending recent results and providing a method that can be refined for even tighter bounds.

Contribution

The authors extend recent bounds to the case where $b eq k$ and strengthen bounds for specific small values of $b$ and $k$, introducing a reduction-based optimization method.

Findings

Extended bounds to $b eq k$ cases.

Strengthened bounds for small $b,k$ values.

Method based on finite case reduction for optimization.

Abstract

For fixed integers $b\geq k$, a problem of relevant interest in computer science and combinatorics is that of determining the asymptotic growth, with $n$, of the largest set for which a $(b, k)$-hash family of $n$ functions exists. Equivalently, determining the asymptotic growth of a largest subset of $\{1,2,\ldots,b\}^n$ such that, for any $k$ distinct elements in the set, there is a coordinate where they all differ. An important asymptotic upper bound for general $b, k$, was derived by Fredman and Koml\'os in the '80s and improved for certain $b\neq k$ by K\"orner and Marton and by Arikan. Only very recently better bounds were derived for the general $b,k$ case by Guruswami and Riazanov while stronger results for small values of $b=k$ were obtained by Arikan, by Dalai, Guruswami and Radhakrishnan and by Costa and Dalai. In this paper, we both show how some of the latter results extend to $b\neq k$ and further strengthen the bounds for some specific small values of $b$ and $k$. The method we use, which depends on the reduction of an optimization problem to a finite number of cases, shows that further results might be obtained by refined arguments at the expense of higher complexity which could be reduced by using more sophisticated and optimized algorithmic approaches.