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?
Decoding a binary linear code given its generator matrix
matrices;coding theory;error correcting codes