A Note on Second-Order Expected Maximum-Load Bounds for Binary Linear Hashing

Abstract: Let $S\subseteq F_2<sup>u$ have size $n=2<sup>\ell$, and let $h:F_2^u\to F_2<sup>\ell$ be a uniformly random linear map. For $y\in F_2<sup>\ell$, write $Load_h(y):=|h^{-1}(y)\cap S|$, and let $M(S,h):=\max_{y\in F_2^\ell} Load_h(y)$ be the maximum load. Jaber, Kumar and Zuckerman (STOC 2025) proved that the expected maximum load of $h$ on $S$ is at most $16\log n/\log\log n$, matching the fully independent keys-into-bins scale up to constants. Their proof also gives the tail estimate [ \Pr\left[ M(S,h)\ge R\frac{\log n}{\log\log n} \right] \le O\left(\frac{1}{R^{2}}\right). ] We record a base optimization in their exponential-potential method showing that binary linear hashing nearly matches fully independent hashing also at the level of the second-order maximum-load scale. For every $R&gt;1$ satisfying $R\ell^{1-1/R}\ge D\ln\ell$, where $D$ is an absolute constant, we prove [ \Pr\left[ M(S,h)\ge R\frac{\log n}{\log\log n} \right] \le O\left( \frac{(\log\log n)^2}{R^2(\log n)^{2-2/R}} \right). ] Integrating this tail yields [ E[M(S,h)] \le \left( 1+ (1+o(1)) \frac{\log\log\log n}{\log\log n} \right) \frac{\log n}{\log\log n}. ] Thus binary linear hashing matches fully independent hashing in the leading term and matches the dominant second-order correction up to a $1+o(1)$ factor. We also prove, by an independent self-contained argument, a sharp tail bound for one prescribed bucket: for fixed $y\in F_2<sup>\ell$, [ \Pr[ Load_h(y)>2^a-2]\le γ^{-1}2^{-a^2}, ] where $γ=\prod_{j\ge1}(1-2^{-j})$. A subspace construction shows that this is asymptotically tight even in the leading constant as $a\to\infty$. However, this controls only a fixed bucket; a direct union bound over all buckets loses a factor $2^\ell$.