The probabilistic bound for perfect hashing is sharp only for binary alphabets

Abstract

We investigate the optimal asymptotic rate $R_q$ of perfect $q$-hash codes, a fundamental extremal problem in combinatorics, information theory, and computer science. Let $B_q=-\log_2(1-q!/q^q)/(q-1)$ denote the classical probabilistic lower bound. We prove that $R_q=B_q$ if and only if $q=2$, establishing an explicit strict improvement for every $q\ge3$ and settling the first assertion of a conjecture of Xing and Yuan (Combinatorica 43 (2023), 347--366). More significantly, we show that, for every $η>0$, $R_q\ge B_q+\exp(-(2+η)q)$ for all sufficiently large integers $q$. This improves the previously known increments of scale $\exp(-3q+O(\ln q))$ by an exponential factor. We also establish a structural limitation of the concatenation method. For pairwise uniform inner codes of subexponential length, the resulting rate bound exceeds $B_q$ by at most $\exp(-3q+o(q))$. Thus, exponentially long inner codes are necessary to achieve the new improvement scale within this framework. Our proofs combine Fourier analysis, high-dimensional simplex codes, additive permutation counts, and entropy bounds for hypergraph matchings. We further strengthen lower bounds from short inner codes and improve several known numerical estimates.