How do I find generator matrix?

Question

Given is matrix:

$\begin{pmatrix} 1 &1 &1 &1 &0 &1 &0 &1 \\ 1&0 & 0 &1 &1 &1 &1 &1 \\ 0&1 &0 &1 &1 &0 &0 &1 \\ 0&0 &1 &0 &1 &1 &1 &0 \\ 0&1 &1 &0 &1 &0 &1 &0 \end{pmatrix}$

Find out if this matrix is valid generator matrix, if not remove some rows of matrix to get a valid generator matrix G.

Solution: Last row (5.) is remowed.

How do I know which row I should remove. I figured out only that adding first two rows we will get the last removed row.

Answer 1

In coding theory, a generator matrix is a matrix whose rows form a basis for a linear code. Hence as long as the rows are linearly dependent and span the original row space, then it is a valid generator matrix. The answer is not unique.

After you remove the $5$-th row, check that the rows are linearly independent, if they are then you are done.

Alternatively, you can also remove the first row and check that the rows are linearly independent.