ISL 1989 Problem 54

Let $n = 2k - 1$ where $k \geq 6$ is an integer. Let $T$ be the set of all $n-$ tuples $(x_1, x_2, \ldots, x_n)$ where $x_i \in \{0,1\}$ $\forall i = \{1,2, \ldots, n\}$ For $x = (x_1, x_2, \ldots, x_n) \in T$ and $y = (y_1, y_2, \ldots, y_n) \in T$ let $d(x,y)$ denote the number of integers $j$ with $1 \leq j \leq n$ such that $x_i \neq x_j$ , in particular $d(x,x) = 0.$ Suppose that there exists a subset $S$ of $T$ with $2^k$ elements that has the following property: Given any element $x \in T,$ there is a unique element $y \in S$ with $d(x, y) \leq 3.$ Prove that $n = 23.$