IrMO 1990 Problem 9

Let $n=2k-1$ , where $k\ge 6$ is an integer. Let $T$ be the set of all $n$ -tuples $$\textbf{x}=(x_1,x_2,\dots ,x_n), \text{ where, for } i=1,2,\dots ,n, \text{ } x_i \text{ is } 0 \text{ or } 1.$$ For $\textbf{x}=(x_1,x_2,\dots ,x_n)$ and $\textbf{y}=(y_1,y_2,\dots ,y_n)$ in $T$ , let $d(\textbf{x},\textbf{y})$ denote the number of integers $j$ with $1\le j\le n$ such that $x_j\neq x_y$ . $($ In particular, $d(\textbf{x},\textbf{x})=0)$ .

Suppose that there exists a subset $S$ of $T$ with $2^k$ elements which has the following property: given any element $\textbf{x}$ in $T$ , there is a unique $\textbf{y}$ in $S$ with $d(\textbf{x},\textbf{y})\le 3$ .

Prove that $n=23$ .