How to find generator matrix from the lexicographic parity-check matrix of a hamming code?
And also how to make it (by row operations) in a way that all columns of weight one are included in it?
For example for $[7,4]$Hamming code with Parity-check matrix: $$ H = \begin{bmatrix} 0 & 0&0&1&1&1&1\\ 0&1&1&0&0&1&1\\ 1&0&1&0&1&0&1 \end{bmatrix} $$ the desired generator matrix is:
$$ G = \begin{bmatrix} 1 & 0& 0 &0 &0 &1 &1\\ 0 &1 &1 &0 &0 &1 &1\\ 0 &0 &1 &1 &0 &0 &1\\ 0 &0 &1 &0 &1 &1 &0\\ \end{bmatrix} $$ where the single-weight columns are columns 1, 2, 4, and 5