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Given polynomial $P(x)=x^6+x^3+1$ belonging to $\mathbb{Z}_2[x]$. Build an $LFSR$ corresponding to $P(x)$. Then find the maximal period of its output sequence and the initial state that could lead to the maximal period output sequence.

Could someone help me?

maximal period=2^n-1 63 if its primitive polynomial initial state=?

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    Search for more questions on LFSRs on this site for examples. Dilip Sarwate has given some IMHO very illuminative answers here.2012-11-05

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Hint: $x^9+1=(x^3+1)(x^6+x^3+1)$ so your feedback polynomial is not primitive.