Average pairwise Hamming distance of a subset in a hypercube

In this note, we prove that for a subset F of \0,1\ⁿ of density 1/8, its average pairwise Hamming distance is at least 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.