$L$ is decidable. Prove that "similar" language is also decidable
Let's consider for simplicity the language
$$
L'' = \{ w : \text{there exists $u \in L$ such that $u,w$ disagree on all positions} \}
$$
instead of your more complicated one.

Consider the language
$$
L = \{ 0^n10^m : \text{the $n$th Turing machine halts on the empty input after exactly $m$ steps} \},
$$
which is computable. You can check that $1^n0 \in L''$ iff the $n$th Turing machine halts, hence $L''$ is not computable.

In order to answer your original question, just repeat each bit 3 times.