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In the CLOSEST STRING problem, one is given (bit-)strings $s_1, \dots, s_t$, each of length $L$ and an integer $d$. The question to be answered is whether there exists a string $s$ that has hamming distance to each $s_i$ that is at most $d$.

In this paper, the authors show that (assuming ETH) the problem cannot be solved in time sub-exponential in $d$, but no requirements on $t$ seem to be placed. However, for $t = 2$, the problem seems to be easy.

Solution for t=2: Let $s_1$ and $s_2$ be the two input strings. Let $k$ be the hamming distance between $s_1$ and $s_2$. If $k/2 \leq d$, then output "yes", otherwise output "no".

Question: Which requirements on $t$ need to be placed for the lower bound in the paper to hold?

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  • $\begingroup$ The problem is even easier for $t=1$ (or for $L=1$, or both, etc.) :-P "The problem cannot be solved in time sub-exponential in $d$" means "There is no way to solve any instance I might come up with of the problem, including an instance with large $t$, in time sub-exponential in $d$". $\endgroup$ Commented Dec 16, 2020 at 3:19
  • $\begingroup$ If you follow the proof in the paper, you will see what value of $t$ their reduction gives. $\endgroup$ Commented Dec 16, 2020 at 8:51
  • $\begingroup$ For any constant $t$, there are only $2^t$ types of indices, so the problem can be solved in $O(n^{2^t-1})$. This can be improved to $O(n^{2^{t-1}-1}+n)$ (recovering your result) by identifying a pattern with its complement. $\endgroup$ Commented Dec 16, 2020 at 8:53

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