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超立方体中子集的平均成对汉明距离

Average pairwise Hamming distance of a subset in a hypercube

Yijia Fang, Hao Huang

AI 速览：作者证明密度为1/8的{0,1}^n子集的平均成对汉明距离至少为(n-3)/2，且等号仅由(n-3)维子立方体达到。登录阅读完整解读 →

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.