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?