Let $R>0$ be given and let $H^n$ be the unit hypercube in $\mathbb{R}^n$. The problem I am facing is to find the maximum number of vertices of $H^n$ which can be covered by a closed $n$-dimensional ball of radius $R$.

I would greatly appreciate any references to literature or ideas on how to approach the problem. I strongly suspect that a solution for any $R$ comes from placing the center of the ball at $(\frac{1}{2},\frac{1}{2},\ldots,\frac{1}{2},0,\ldots,0)$ with the number of nonzero coordinates being $\lfloor 4R^2 \rfloor=\lfloor D^2 \rfloor $, where $D$ is the diameter (for $D \leq n$).

Any references or ideas are welcome.

Thanks!