Given a prime $p$, how many vectors of length $p^k$ over $\Bbb F_2$ of Hamming weight $c^k$ can one have such that the Hamming distance between any pair of vectors is $2(c^k-c^{k-1})$ 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?