I am a little lost in regards to handle this question. I am aware that a cyclic code of size $[n=8,k=3]$ must have a generator polynomial $g(x)$ of degree $n-k$ which equates to $8-3=5$ in this case. Also it must be a factor of $x^8+1$.
We are working in GF(2) however so I use maple to factor and gain:
$1+x^8 = (1+x^4)^2=(1+x)^8$.
Niether of these are of degree $5$ so I feel as though I may have misunderstood something.
I think the generator matrix should be:
$\begin{bmatrix}1 & 1 & 0 & 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 \end{bmatrix} $
But this is based on finding $g(x)=1+x+x^3$. I'm not sure if that is what I should be finding here and if so, how to go about it.
Any help is much appreciated.