Given a positive integer $n$, the Hamming distance $d^H_n(x,y)$ of $x,y\in \{0,1\}^n$ is defined by $$d^H_n(x,y) = |\{k\in\{0,\ldots,n-1\}: x(k)\neq y(k)\}|.$$

Given an integer $n>0$ and a set $S\subseteq \{0,1\}^n$ with $|S| = n$, is it possible to find a map $f:S\to \{0,1\}^{n+1}$ such that $$d^H_{n+1}(f(x), f(y)) = d^H_n(x,y) + 1 \text{ for all } x\neq y\in S$$?