Let $X=\{0,...,2^n-1\}$. Is there a $Y\subseteq X$ with $|Y|=2^{n-1}$ and such that $d(x,y)\geq 2$ for any $x,y\in Y$, where $d$ is the hamming distance (number of on bits of the xor operation)?
I know this question is more general, but I am only interested in the specific case of $d_\min=2$, for which according to the link provided in one of the answers, such a $Y$ exists.