"Find a linear $1$-error-correcting code $C$ of length $12$ satisfying the following conditions or disprove its existence.
(1) The number of words of $C$ is $256$.
(2) The word $c = 111110000011$ belongs to $C$.
(3) The word $c' = 000111000101$ does not belong to $C$ but can be corrected to a word of $C$."
I know that, if such a code exists, then the distance between any two codewords must be at least $3$. I also see that the distance between $c$ and $c'$ is $6$, and since the code is $1$-error-correcting, $c'$ contains exactly $1$ error. But I'm not sure how to make use of condition (1). Any hints?