Given a positive integer $n$, the Hamming distance $d^H_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$ We say that a positive integer $s$ is $n$-spreadable if there is $T\subseteq \{0,1\}^n$ with $|T|=n$ and for $x\neq y\in T$ we have $d_H(x,y) \geq s$. For any integer $n\geq 1$ let $m_n$ be the largest $n$-spreadable number less or equal to $n$.

Question. Do we have $\lim \sup_{n\to\infty}\frac{m_n}{n} = 1$?