How to find the linear codes of a linear generator matrix? This is the last matrix generator from here
https://proofwiki.org/wiki/Standard_Generator_Matrix_for_Linear_Code/Examples#:~:text=1%20transmission%20error.-,Linear%20(4%2C2)%2Dcode,in%20Z3%3A%20Example%201&text=G%3A%3D(102,and%20corrects%200%20transmission%20errors.
Let $G=\begin{pmatrix}
1& 0 & 1 & 1\\
0 & 1 & 1 & 2
\end{pmatrix}$ be the standard generator matrix. Then $G$ generates the linear code $C=\{0000,0112,0221,1011,1120,1202,2022,2101,2210\}$.
I don't understand how $0221$ is an element of $C$. It is given by $0*(1001)+2*(0112)$. But as I see it, it should equal $0220$ and not $0221$. So I perform vector addiction and scalar multiplication as normal(vectorspace) and then I take mod 2 of the result, but that obviously isn't the right way.

The example is over $\mathbb{Z}/3\mathbb{Z}$, so any modular reductions should be modulo 3!
