# LFSR sequences over GF(4)

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.

Is this a homework?

No. I am studying sequences over extension fields. I need make with a program. I tryed to make on SAGE but I could not. If I can define a recurring sequence over GF(4) then I can apply to other examples.

Check out documentation and examples at https://doc.sagemath.org/html/en/refe...

I checked this document page. The elements of F=GF(4) are (0, 1, 0, 1) in order. I dont get it and asked for help here.

Please add to your question a Sage code you have trouble with, along with the expected outcome.