Feedback-polynomial parity condition for maximal period
= Feedback-polynomial parity condition for maximal period
A maximal-period feedback polynomial cannot have $1$ as a root. Over $\mathbb F_2$, this says
$$
1+a_{d-1}+\cdots+a_0\ne0,
$$
so an even number of the coefficients $a_i$ equal one and the full monic polynomial has an odd number of nonzero coefficients.