I want to prove the following theorem: Let's assume a LFSR(Linear feedback shift register) sequence with a reducible characteristic polynomial of degree $n$ over the finite field $\mathbb{F}_q$. Under those assumptions, the minimal period of this sequence cannot be $q^nā1$ (which implies it must be smaller).
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This might help you. https://math.stackexchange.com/questions/872984/sequences-length-for-lfsr-when-polynomial-is-reducible Also See the Golombs Book. ā kelalaka Apr 14 '20 at 12:50