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

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