Let $C$ be the [5,4] code over $F_7$ with generator matrix $G=$
$$\begin{pmatrix} 1 & 0 & 3 & 5 & 4 \\ 0 & 0 & 2 & 3 & 5\\ 2 & 1 & 0 & 3 & 0\\ 1 & 1 & 0 & 0 & 0\\ \end{pmatrix}$$
Find a generator matrix $G$ for $C$ in standard form using only row operations
Start of solution:
$G$ = $$\begin{pmatrix} 1 & 0 & 3 & 5 & 4 \\ 0 & 0 & 2 & 3 & 5\\ 2 & 1 & 0 & 3 & 0\\ 1 & 1 & 0 & 0 & 0\\ \end{pmatrix}$$
$ \rightarrow$
$$\begin{pmatrix} 1 & 0 & 3 & 5 & 4 \\ 0 & 0 & 2 & 3 & 5\\ 0 & 1 & 1 & 0 & 6\\ 0 & 1 & 4 & 2 & 3\\ \end{pmatrix}$$
So I am confused about how they get say 6 on row 3 in the 4th column.
I know how to do elementary row operations and gaussian elimination etc but this is part of a coding theory course and I think something different to normal row operations is being used, as I cannot see any way of getting the next stage matrix above.
Please could someone explain how the above has been calculated?
Many thanks
The usual business of putting more zeros on the first column. Multiples of the first row were subtracted from the two last rows. In particular the new fourth row is gotten from the old one by subtracting the first row multiplies by two from it. All the arithmetic is done modulo $7$, so one entry at a time it goes like $$ \begin{aligned} 2-2\cdot1&=0,\\ 1-2\cdot0&=1,\\ 0-2\cdot3=-6&\equiv1,\\ 3-2\cdot5=-7&\equiv0,\\ 0-2\cdot4=-8&\equiv6. \end{aligned} $$