My goal is to construct a generator matrix for a classical binary $[8,2,5]$ Goppa code $\Gamma(L,G)$, with $L=\mathbb{F}_8$ and Goppa polynomial $G(x)=x^2+x+1$.
So far I have successfully been able to construct the parity check matrix $H$.
$$ H = \begin{bmatrix}
1 & 1 & \alpha^2 & \alpha^4 & \alpha^2 & \alpha & \alpha & \alpha^4 \\
0 & 1 & \alpha^3 & \alpha^6 & \alpha^5 & \alpha^3 & \alpha^6 & \alpha^3
\end{bmatrix}$$ $$= \begin{bmatrix}
1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 \\
0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 \\
0 & 0 & 1 & 0 & 1 & 0 & 0 & 1 \\
0 & 0 & 0 & 1 & 1 & 1 & 1 & 0
\end{bmatrix}$$
From class I know that I can find the generator matrix through finding the kernel of H. However I am quite stuck on how to do this. Any help would be greatly appreciated. Thanks in advance!
Best Answer
Yes, you can find the generator matrix through finding the kernel of $H$, but you can find it through writing $H$ in its standard form, i.e., $H = \left[ A | I_{n \times k}\right]$. Then you use $A$ to build your generator matrix (in its standard form): $G = \left[ I_k | A^t \right]$, where $A^t$ is the transpose of the matrix $A$.
I am working with the same Goppa code and I got this generator matrix:
$$G = \begin{bmatrix} 1 & 1 & 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 1 & 1 \end{bmatrix}$$