Consider the set of all $k$-subsets of $\{1,\dots,n\}$, naturally identified with a subset $A$ of $\{0,1\}^n$ where each element has exactly $k$ ones. Is there a sharp bound known for $\epsilon$-covering of this set in the Hamming distance?

More specifically, suppose that $k = \gamma n$ where $\gamma \in(0,1/2)$ is fixed. The cardinality of $A$ is asymptotically $|A| \sim e^{n h(\gamma)}$ where $h(\cdot)$ is the binary entropy function. Is there an $\epsilon$-covering of $A$ in Hamming distance with $e^{ \frac{C n}{\log n}}$ elements and say $\epsilon \le \frac{n}{(\log n)^{2}}$? In other words, how large $\epsilon$ needs to be to be able to go from cardinality being exponential in $n$ to it being exponential in $n / \log n$.