What is the method to find a generator matrix and a check matrix of a Hamming code?
I'm current trying to find a generator matrix of Ham$(2,3$).
This is how far I have got:
$$n = \frac{q^s - 1}{q - 1} = \frac{3^2 - 1}{3-1} = 4$$
Therefore there are 4 columns and 2 rows of the generator matrix.
So the generator matrix will take the form of \begin{bmatrix} 1 & 0 & ? & ? \\ 0 & 1 & ? & ?\end{bmatrix}
How do I know what the $2 \times 2$ matrix to the right should contain?