Lower limit on the number of check bits needed to correct single errors
The comment by @gnasher is right. Moreover, if the code was $k$ error correcting with $m$ data bits, each of the $2^m$ legal messages would require
$$
1+n+\binom{n}{2}+\cdot+\binom{n}{k}=\sum_{j=0}^k \binom{n}{j}
$$
bit patterns dedicated to it, where $\binom{n}{j}$ is the number of words at Hamming distance $j$ from a given word.