Lets say I have $C=\{00000, 01101, 10110, 11011\}$
I know the covering radii is given by $e=\frac{1}{2} (d-1)$ So in this example $e=1$.
I know the covering radius of a code C in $(F_q)^n$ is the smallest integer $\rho=\rho(C)$ such that
$\cup_{x \in C}$ $ S(x,\rho)=(F_q)^n$
What is this definition trying to say, how can we interpret it?
In my example, why is the covering radius $\rho=2$?
How do we get this answer?
Take the null word:
(00000)
If you generate a list of all words that have a distance of 2 to this word, you get:
(00011) (00101) (00110) (01001) (01010) (01100) (10001) (10010) (10100) (11000)
If you do that with all your codewords, you get 4 sets with 10 words each. Some words are included in several sets.
The complete vector space contains $5^2 = 32$ words. If what you say is true and $p = 2$, then these words are all included in the unification of your 4 sets. This is the meaning of the covering radius. If you do the same with a radius of 1, the complete vector space will not be a subset of the unification of your 4 sets.