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arXiv · 2610.40486

Average pairwise Hamming distance of a subset in a hypercube

Yijia Fang·Hao Huang
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Abstract

In this note, we prove that for a subset $\mathcal{F}$ of $\{0,1\}^n$ of density $\frac{1}{8}$, its average pairwise Hamming distance is at least $\frac{n-3}{2}$, with equality attained only by $(n-3)$-dimensional subcubes. This answers a question of Lei Yu, and solves an important open case of a more general question posed by Ahlswede and Katona in 1977.

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Yijia Fang, Hao Huang. 2026-10-29. Average pairwise Hamming distance of a subset in a hypercube. https://arxiv.org/abs/2610.40486

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