So I was given the following (6,3) block code with some unknown values marked with x:
format: index,(msg) -> (code word)
1 (0 0 0) -> (0 0 0 0 0 0)
2 (1 0 0) -> (0 1 1 1 0 0)
3 (0 1 0) -> (1 1 1 0 1 0)
4 (1 1 0) -> (1 0 0 1 1 0)
5 (0 0 1) -> (1 1 0 0 0 1)
6 (1 0 1) -> (1 0 1 1 0 1)
7 (0 1 1) -> (0 0 0 0 1 1)
8 (1 1 1) -> (x x x 1 1 1)
The goal is to find the generator matrix. My first idea is to take the vectors of code words 2, 3 and 5 and form the generator matrix.
$G = \left[ \begin{array}{rrr} 0&1&1&1&0&0 \\\ 1&1&1&0&1&0 \\\ 1&1&0&0&0&1 \end{array} \right]$
Now to check if this is correct, I would calculate the code words for (1 1 0), (1 0 1), (0 1 1) and find the missing bits for code word (1 1 1).
(1 1 0) * G = (1 0 0 1 1 0)
(1 0 1) * G = (1 0 1 1 0 1)
and (0 1 1) * G = (0 0 1 0 1 1), but according to the table, this is not the right code word. Where did I make a mistake?