I do not understand how to determine a generator matrix. I have attached a question below. Can someone please explain to me how to do this question?
The vectors $C_0=[10001]$, $C_1=[11010]$, and $C_2=[11101]$ form a basis for a $(5,3)$ block code $C$ over $\text{GF}(2)$.
Use $C_0$, $C_1$, and $C_2$ to construct a generator matrix $G$, and then transform $G$ into systematic form $\tilde{G}$.
Well, a generator matrix contains a basis of the code as row vectors:
$C = \begin{pmatrix} 1&0&0&0&1\\ 1& 1&0&1&0\\ 1&1&1&0&1 \end{pmatrix}$
Now you can perform row operations to form a systematic code.