Given a prime $p$, out of $N$ vectors of length $p^k$ over $\Bbb F_2$ of Hamming weight $(\frac{c+1}{2})^{k}$ that are chosen, how many vectors can there be with pairwise Hamming distance at least $2(\frac{c+1}{2})^{k}$ given that for every vector there are exactly $c^k-1$ vectors that are less than Hamming distance $2(\frac{c+1}{2})^{k}$ where $c \geq 3$ is odd?

Has this problem been studied and are there good tools to study this problem for tight upper and lower bounds?