## Error Correcting Code

Theorem. The code $C$ is $e$-error correcting if and only if its minimum distance is $2e+1$ or greater.

For the $(\Rightarrow)$ direction we suppose for contradiction that the minimum distance is $d \leq 2e$. So then $d/2 \leq e$. Now $[d/2] \leq d/2$. Also $d-d/2 = d/2 \leq e$. But from this we cannot conclude that $d- [d/2] \leq e$ right?