Let $k, n \in \mathbb{N}$, $k < (1 - \epsilon)n$ for some little $\epsilon > 0$. I want to find $f: \{0,1\}^k \to \{0, 1\}^n$ such that:

1. $f(a) \not= f(b)$ if $a \not=b $

2. for any $x \in \{0,1\}^n$ $V_x(k/2) \cap Im(f) \le 2^{k(1 - \delta)}$ for some $\delta > 0$, where $V_x(k/2)$ is a full-sphere with center $x$ and radius $k/2$ (in Hamming's metric ).

3. $f$ can be calculated fast - in polynomial(k) time

Whether there is such $f$?