I want to know how to apply definition of perfect code to this code ( I want to prove $C$ is perfect :-
The binary codes $C$ of odd length consisting of a vector $ \bf{c} $ and the complementary vector $\bar{\bf{c}} $ with $0$s and $1$s interchanged .
Hint For any binary word $w$ of length $2n+1$, (where $2n+1$ is your odd number) you can prove that
$$d(w,c)+d(w,\bar{c})= 2n+1 \,.$$
This is easy because every binary digit in $w$ is either the same as the digit in $c$ and opposite than the digit in $\bar{C}$, or the other way around.
Now, all you need is show that if two non-negative integers add to $2n+1$, one has to be $\leq n$ and the other has to be $\geq n+1$.