As a (somewhat disguised) extension of my recent MO post.

Given integers $n\ge k\ge 1$ and $N$, for a collection of $N$ Hamming spheres in ${\mathbb F}_2^n$ of radius $1$, what is the smallest possible number of points in ${\mathbb F}_2^n$ covered by at least $k$ spheres?

Suppose that all our spheres are centered at the even-weight points of ${\mathbb F}_2^n$, and denote by $F_0$ the set of all even-weight points. (The motivation here is that even-centered and odd-centered spheres are disjoint; hence, the sets they $k$-cover are disjoint, too.) It is not difficult to find a linear subspace $L<F_0$ of co-dimension $\lceil\log_2n/(k-1)\rceil$ such that no point in ${\mathbb F}_2^n$ is covered by $k$ (or more) spheres, centered at the elements of $L$. Therefore, if we have fewer than $\sim2^{n-1}k/n$ spheres, then the set of $k$-covered points can be empty. On the other hand, by a simple double counting, for any system of $N$ even-centered spheres, the number of $k$-covered points is at least $$ \Big(1-\frac{2^{n-1}k}{n}\Big)\,N. $$ My question is: assuming that $k$ is reasonably small, and $N$ is reasonably large, can this estimate be improved to show that the number of $k$-covered points is at least $N$? Without going into details, I would consider "reasonable" the values $k<n^\epsilon$ and $N\ge2^{n-2}$ (although I have no evidence that the estimate in question fails slightly outside this range). And so, to put it in a self-contained form:

Is it true that for any $k\le n^\epsilon$ and any collection of $N\ge 2^{n-2}$ even-centered unit spheres in ${\mathbb F}_2^n$, there are at least $N$ points, covered by $k$ or more spheres?

The answer is (trivially) positive for $k=1$, and the case $k=2$ is treated here. Unfortunately, the argument for the case $k=2$ does not seem to extend onto the larger values of $k$.

A curious observation is that if $C$ and $\overline C$ are complementary subsets of $F_0$, and $S$ is the set of all points $k$-covered by the spheres centered at the elements of $C$, then $\overline S:=F_0\setminus S$ is exactly the set of all points $(n-k)$-covered by the spheres centered at the elements of $\overline C$. This shows that my question can be equivalently restated as follows:

Is it true that for any $k\le n^\epsilon$ and any collection of $N\le 2^{n-2}$ even-centered unit spheres in ${\mathbb F}_2^n$, there are at most $N$ points, covered by $n-k$ or more spheres?