What is the maximum number of pairs of $k\geq2$ distinct binary strings of length $n$, such that the strings in each pair are at Hamming distance $b$? In other words, given $k$ points on the $n$-dimensional Boolean hypercube, how many of these can possibly be at distance $b$ of each other?
I have a trivial upper bound given by $k \min\{\binom{n}{b},k\}$ simply because each string can have at most $\binom{n}{b}$ neighbors at distance $b$. Does anybody know how to better upper bound this quantity?