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

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

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