Question

Suppose we build a Linear Feedback Shift Register machine that works mod 3 instead of mod 2. It uses a recurrence of length 2 of the form
xn+2 ≡ c0xn + c1xn+1 (mod 3) to generate the sequence 1, 1, 0, 2, 2, 0, 1, 1.
Set up and solve the matrix equation to find the coefficients c0 and c1.

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Because the operations are performed mod 3, the coefficients can be 0, 1 or 2.

We need only two equations to set the matrix (the coefficients can be also found by solving two recurrence relations as well).

The sequence is x0,x1, x2, x3…. is 1,1,0,2,2,0,1,1.
This means we need to plug into the provided recurrence relations the following:
1,1 |0 and 1,0|2
1,1|0 => x2=c0*x0+c1*x1 (mod 3)=> c0*1+c1*1=0 (mod 3)
1,0 |2 => x3=c0*x1+c1*x2 (mod 3)=> c0*1+c1*0=2 (mod 3)...

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