Assuming I have the following code: $C = {00000,01001,01110,00111}$
My task is to get the parity check matrix $H$ for this code. So I created the generator matrix for this code, which is:
$G = \begin{bmatrix} 0 & 1 & 0 & 0 & 1 \\ 0 & 1 & 1 & 1 & 0 \\ \end{bmatrix} $
In class, I learned that the conversion between the generator matrix and the parity check matrix works in the following way:
$G = [I_k|A]$
$H = [A^T|I_{n-k}]$
However, in this case my generator matrix does not have the identity matrix on the left, and I cannot generate the generator matrix by multiplying a value or adding the two rows together, as none of these actions would change the first digit I would need to change in order to get an identity matrix.
How do I proceed here in order to get the parity check matrix for the given code?