I want to create sequences generated by the polynomial $x^2+\alpha x+1$ where coefficients are from $\mathbb{F}_4$ s.t. $\alpha^2 + \alpha +1=0$. In which $\alpha^2=\alpha+1$ and $\alpha^3=1$.
There are 16 initial states which are:
00,01,0$\alpha$,0$\alpha^2$
10, 11, 1$\alpha$, 1$\alpha^2$
$\alpha$0, $\alpha$1, $\alpha$$\alpha^2$, $\alpha$$\alpha^2$
$\alpha^2$0, $\alpha^2$1, $\alpha^2$$\alpha$, $\alpha^2$$\alpha^2$
For example, by starting with initial state $\alpha$$\alpha^2$, we get 1$\alpha$ + $\alpha$$\alpha^2$= $\alpha$+$\alpha^3$= $\alpha$+1= $\alpha^2$.
Now the sequence is $\alpha \alpha^2 \alpha^2$. To find the 4. item we will use the new state $\alpha^2 \alpha^2$ and we get 1$\alpha^2$ + $\alpha$$\alpha^2$= $\alpha^2$+$\alpha^3$= $\alpha^2$+1= $\alpha$.
Now the sequence is $\alpha \alpha^2 \alpha^2\alpha$.
By the same way we get the sequence $\alpha \alpha^2 \alpha^2\alpha 0 \alpha \alpha^2 \alpha^2...$ of period 5. I could not write the sage code to create the sequence.
