Let $n$ be an odd integer greater than $11$; $k\in \mathbb{N}$, $k \geq 6$, $n=2k-1$. We define \[d(x,y) = \left | \{ i\in \{1,2,\dots, n \} \bigm | x_i \neq y_i \} \right |\] for $T=\{ (x_1, x_2, \dots, x_n) \bigm | x_i \in \{0,1\}, i=1,2,\dots, n \}$ and $x=(x_1,x_2,\dots, x_n), y=(y_1, y_2, \dots, y_n) \in T$. Show that $n=23$ if $T$ has a subset $S$ satisfying $|S|=2^k$ For each $x \in T$, there exists exacly one $y\in S$ such that $d(x,y)\leq 3$