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