Is the language $L=\{\langle M, i\rangle: M$ is a TM, $i \in \mathbb{N}$ and $M$ accepts some string $w\in \{0,1\}^*$ of length $|w|=i.\}$ decidable? I think it's decidable because I can construct a decider $D$. First, enumerate all strings of length $i$. Since $i$ is a fixed natural number, there are $2^i$ binary strings of length $i$. Run $M$ on each string and if $M$ accepts any of them, $D$ accepts $\langle M, i\rangle$. If $M$ rejects all of them, $D$ rejects. I think it's decidable because we can construct a decider and test it on finitely many strings since $i$ is a fixed natural number. It will always halt because it will either accept if any string is accepted or reject if all of them are rejected.
However, I'm not confident with my answer after reading another question on CS SE. Can someone help me clarify?
