Let $C$ be the binary linear code with the following generator matrix
$G= \begin{bmatrix} 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 \end{bmatrix}$
I need to decode the received word $r$, $\begin{pmatrix} 1 & 1 & 0 &0 & 0 &1 &1\end{pmatrix}$.
I am really strugggling with this. I know a message $m$ is encoded as $mG$.
So $mG$=$\begin{pmatrix} 1 & 1 & 0 &0 & 0 &1 &1\end{pmatrix}$.
I have tried doing this by eye but couldn't find a single $m$ that I could multiply $G$ by to give me $r.$ Is there another method? Also I know that there must be $2^4=16$ codewords and when I wrote them out this one wasn't in them? Not sure if that helps or maybe I did that wrong?