Let $S = \lbrace s_1, s_2 \ldots s_n \rbrace$ be a set of strings each of length $k$ from an alphabet $\Sigma$, $h(s_i, s_j)$ denote the hamming distance between two strings. The simultaneous hamming neighborhood is defined as $N_{\alpha} = \lbrace s' | h(s',s_j) \leq \alpha, \forall s_j \in S , s' \in \Sigma^k \rbrace$, $1\leq \alpha \leq k$.

I would like to know if this problem (i.e. computing $N_{\alpha}$ efficiently) has been considered earlier ? -- By efficiently I mean the running time of the algorithm should be something like $O(|N_{\alpha}|)$, when $|N_{\alpha}|$ is much larger than $n$.

Thank you very much for your help.