IR-MO 2004 Problem 3

Suppose $V= \mathbb{Z}_2^n$ and for a vector $x=(x_1,..x_n)$ in $V$ and permutation $\sigma$ .We have $x_{\sigma}=(x_{\sigma(1)},...,x_{\sigma(n)})$ Suppose $n=4k+2,4k+3$ and $f:V \to V$ is injective and if $x$ and $y$ differ in more than $n/2$ places then $f(x)$ and $f(y)$ differ in more than $n/2$ places. Prove there exist permutaion $\sigma$ and vector $v$ that $f(x)=x_{\sigma}+v$