I'm trying to solve the following problem:
$C$ is a binary cyclic $[7,4]$ code with generator matrix
$$ \begin{pmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 0 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} $$
Write down a generating polynomial for $C$ and find a check polynomial $h$.
I find that the generating polynomial is $1 + u + u^{3}$
However, I am stuck on how to work out the check polynomial.
Perhaps $(1 + u + u^{3} ) ( 1 + au + bu^{2} + cu^{3} + du^{4}) = 0$
But how would one solve this? Or can I do long division, is fo can someone please show me how? \thanks