Let $d(v,w)$ denote the Hamming distance of binary strings $v$ and $w$ of the same length. Given binary strings $x_1,\dots,x_k$ of length $n$ such that $d(x_i,x_j)=2\alpha$ for $i\neq j$, how many strings $x$ are there such that $d(x,x_i)=l$ for $i=1,\dots,k$? I can count the number of such strings for the case $k=2$. I am considering the general case $k\geq 3$. I wonder if this kind of question is well-known so that there have already been results.
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(Case $k=2$) WLOG, $x_1=0\dots 0$ and $x_2=0\dots01\dots 1$ where $1$ appears $2\alpha$ times in $x_2$. If $d(x,x_1)=i$, then $x$ must contain $i$ ones. Denote by $S_i(\{x_1,x_2\})$ the set of all binary strings at the same distance $i$ from $x_1$ and $x_2$. It can be checked that $|S_i(\{x_1,x_2\})|=\binom{2\alpha}{\alpha}\binom{n-2\alpha}{i-\alpha}$ for $i=\alpha,\dots, n-\alpha$.