New upper bounds for $(b,k)$-hashing

TL;DR

This paper advances the understanding of upper bounds for the size of perfect $(b,k)$-hashing sets, extending recent bounds to more general cases and improving bounds for specific small parameters.

Contribution

It extends recent bounds for $(b,k)$-hashing to cases where $b eq k$ and improves bounds for certain small $b,k$ values using a reduction-based method.

Findings

Extended bounds to $b eq k$ cases.

Improved bounds for specific small $b,k$ values.

Method based on reduction to finite cases for optimization.

Abstract

For fixed integers $b\geq k$, the problem of perfect $(b,k)$-hashing asks for the asymptotic growth of largest subsets 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.