# 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.

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1

Is this a homework?

( 2023-02-20 14:06:18 +0200 )edit

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.

( 2023-02-20 14:20:57 +0200 )edit
1

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

( 2023-02-20 15:42:41 +0200 )edit

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.

( 2023-02-20 15:54:40 +0200 )edit
1

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

( 2023-02-20 16:44:23 +0200 )edit

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Is this what you want?

K.<a> = GF(4, modulus=[1,1,1])
L = lfsr_sequence([1,a], [0,1], 20)
print(L)

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Thanks a lot dear Max Alekseyev

( 2023-02-21 16:36:09 +0200 )edit