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Given a set $S\subseteq \{0,1\}^d$ of the Boolean hypercube of dimension $d$, define the average distance of $S$ as $$ \bar{d}(S) = \frac{1}{\lvert S\rvert^2} \sum_{x,y\in S} d_H(x,y)\tag{1} $$ where $d_H$ denotes the Hamming distance. For any $1\leq n \leq 2^d$, define the minimum average distance as $$ \beta(d,n) = \min_{\substack{S\subseteq \{0,1\}^d\\ \lvert S\rvert = n}} \bar{d}(S)\tag{2} $$

In [1], Ahlswede and Katona asked $\beta(d,n)$ for arbitrary $n$. Ahlswede and Althöfer, in [2], provided asymptotically tight (in $d$) lower bounds on $\beta(d,n)$, when $n \geq\binom{d}{\alpha d}$ for constant $\alpha \in(0,1/2)$: $$ \liminf_{d\to\infty} \frac{\beta(d,n)}{d} \geq 2\alpha(1-\alpha)\tag{3} $$

In [3], Althofer and Sillke also showed the general bound $$ \beta(d,n) \geq \frac{1}{4}\left(d+1-\frac{2^d}{n}\right) \tag{4} $$

Besides that, there is some work showing optimal bounds for some cases of the form $n= 2^{d-1}\pm O(1)$ (see e.g. [4]), and it seems most of the more recent work is focused on those, and the structure of optimal large sets achieving these bounds.

So it looks like, for $n= 2^{d-k}$, the cases $k=\Theta(d)$ (via (3)) and $k=O(1)$ are rather well-understood; but for the general case, (4) only gives $$ \beta(d,n) \geq \frac{1}{4}\left(d+1-2^k\right) \tag{5} $$ while the other cases seem to hint at a behavior more like $\beta(d,n) \approx d-k$.

Is there anything better than (4) known for the "intermediary regime" where $n=2^{d-k}$ with $1 \ll k \ll d$ (e.g., $k = \log d$, or $k = \sqrt{d}$)?


[1] R. Ahlswede and G.O.H. Katona, “Contributions to the geometry of hamming spaces”, Discrete Mathematics, vol. 17, 1977, pp. 1-22.

[2] R. Ahlswede and I. Althöfer, “The asymptotic behavior of diameters in the average”, Journal of Combinatorial Theory, Series B, vol. 61, 1994, pp. 167-177.

[3] Ingo Althöfer, Torsten Sillke, 1992, 'An “average distance” inequality for large subsets of the cube', Journal of Combinatorial Theory, Series B, vol. 56, no. 2, pp. 296-301

[4] André Kündgen, 2002, 'Minimum average distance subsets in the hamming cube', Discrete Mathematics, vol. 249, no. 1-3, pp. 149-165

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