New bounds for perfect $k$-hashing

TL;DR

This paper improves bounds on the size of perfect k-hashing sets by proving a conjecture and developing new methods, leading to tighter bounds for all k and significant improvements for k=5,6.

Contribution

The paper proves a conjecture on polynomial maxima and introduces a new method that improves bounds for perfect k-hashing, especially for k=5,6.

Findings

Proved the conjecture on polynomial maxima, completing explicit bounds.

Developed a new method that improves bounds for k=5,6.

Established tighter bounds for all k in perfect k-hashing.

Abstract

Let $C\subseteq \{1,\ldots,k\}^n$ be such that for any $k$ distinct elements of $C$ there exists a coordinate where they all differ simultaneously. Fredman and Koml\'os studied upper and lower bounds on the largest cardinality of such a set $C$, in particular proving that as $n\to\infty$, $|C|\leq \exp(n k!/k^{k-1}+o(n))$. Improvements over this result where first derived by different authors for $k=4$. More recently, Guruswami and Riazanov showed that the coefficient $k!/k^{k-1}$ is certainly not tight for any $k>3$, although they could only determine explicit improvements for $k=5,6$. For larger $k$, their method gives numerical values modulo a conjecture on the maxima of certain polynomials. In this paper, we first prove their conjecture, completing the explicit computation of an improvement over the Fredman-Koml\'os bound for any $k$. Then, we develop a different method which gives substantial improvements for $k=5,6$.