I am trying to find the covering radius of $C=\{0000,1111,2222\}$
I know the definition of the covering radius $S$ for a code $C$ is the smallest non negative integer y such that $$\cup_{x \in C} S(x,y)=V(n,q)$$
I dont really understand this definition. I am told the answer is $2$ but I do not know how to get to it.
We want to find the smallest integer $S$ such that for all $x \in F_3^4$, there exists $c \in C$ such that $d(x,c) \leq S$, where $d(x,y)$ is the Hamming Distance of the vectors $x$ and $y$. i.e. every vector in $F_3^4$ is within a radius $S$ of a codeword.
Let $x = (x_0, x_1, x_2, x_3)$. If $x_0 = x_1$, let $c = (x_0,x_0,x_0,x_0) \in C$. Then $d(x,c) \leq 2$.
Suppose $x_0 \ne x_1$. If $x_2 = x_0$ or $x_2 = x_1$, then by a similar argument there exists a codeword in $C$ with distance at most $2$ from $x$. Suppose $x_2 \ne x_0$ and $x_2 \ne x_1$. Then since there are only three distinct elements in $GF(3)$, $x_3$ must equal one of $x_0,x_1,x_2$, so by a similar argument there exists a codeword with distance $2$.
Hence $S \leq 2$. The vector $(0,0,1,1)$ has distance $2$ at least two from all codewords, so $S \geq 2$.
Hence $S = 2$.