Find the order of a polynomial

Question

I want to find the cycle set for the polynomial $p(x)=x^{23}+x^6+1$ over $\mathbb{F}_2$.

So, I have the connection polynomial $C(D)=1+D^{17}+D^{23}$ over $\mathbb{F}_2$

The factors to $C(D)$ are:

1. $C_1(D)=1+D+D^4$
2. $C_2(D)=1+D^2+D^3$
3. $C_3(D)=1+D+ D^3+ D^4 +D^8 + D^{10} +D^{12} +D^{13} +D^{14} +D^{15} +D^{16}$

$C_1$ and $C_2$ are primitive so they are fine.

But I need the period/order of $C_3(D)$.

Anyone who knows how to solve this? Preferably explanation with Maple commands.

Answer 1

C3(D) is irreducible

If C(D) is a primitive polynomial of degree L over q then the cycle set is 1(1) ⊕ (1)(qL − 1).

If C(D) is an irreducible polynomial of degree L over q then the cycle set is 1(1) ⊕ ((q^L - 1)/T)(T)

where T is the period of the polynomial C(D) (or the order of α when π(α) = 0).

For your example q=2, L=4,3,16 , T=21845

Ans

[1(1) ⊕ 1(2^4 - 1)]⨂[1(1) ⊕ 1(2^3-1)]⨂[1(1) ⊕ ((2^16-1)/21845)(21845)]

Final Ans:

1(1)⊕1(7)⊕1(15)⊕1(105)⊕3(21845)⊕15(65535)⊕3(152915)⊕15(458745)