In order to determine the generator polynomial, I should first determine its degree. How do I find the correct degree of generator polynomial $g(x)$ for a cyclic code? The code has length $n = 8$ over $GF(3)$.
I know that $deg(g(x)) = n - k$, with $k$ the dimension, but $k$ is not given.
What I did so far:
$n=8$, $3^{k}$ $($mod $ 8) = 1$. $k$ little as possible $\rightarrow k=2$.
$\beta=\alpha^{(9-1)/8}=\alpha$ with $\alpha$ a primitive element of $GF(9)$:
$\alpha$ root of $x^2+x+2 \rightarrow \alpha^2=1+2\alpha$.
Cyclotomic classes and minimal polynomials:
$C_0=\{0\}\rightarrow 2+x \\ C_1=\{1,3\}\rightarrow 2+x+x^2 \\ C_2=\{2,6\}\rightarrow 1+x^2 \\ C_4=\{4\}\rightarrow 1+x \\ C_5=\{5,7\}\rightarrow 2+2x+x^2 \\ \rightarrow x^8-1=(2+x)(2+x+x^2)(1+x^2)(1+x)(2+2x+x^2)$.
I know the properties of a generator polynomial $g(x)$ but don't know how to apply them to determine the degree.