I have a $\mathbb{F}_2^{n\times n}$ matrix $P$ = $ \begin{bmatrix} 0 & 0 & \ldots & 0 & -c_0 \\ 1 & 0 & \ldots & 0 & -c_1 \\ 0 & 1 & \ldots & 0 & -c_2 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ 0 & 0 & \ldots & 1 & -c_{n-1} \end{bmatrix} $ that has characteristic polynomial $p(x) = x^n + c_{n-1}x^{n-1} + \ldots + c_1 x + c_0$ such as roots of $p(x)$ are primitive $\mathbb{F}_{2^n}^*$ elements.
I can write $P = QDQ^{-1}$ where $Q$ is the basis of eigenvectors and $D$ is the diagonal matrix of eigenvalues.
I am looking for the set of matrices $P' = QD'Q^{-1} $such that $ D'$ is diagonal, its coefficients are primitive $\mathbb{F}_{2^n}^*$ elements, and P' elements are all ones or zeroes
Is P the only one in this set ? If not,
a) can I construct such $P'$ matrix ?
b) How many such $P'$ matrices exist ?
c) Can I sample a random matrix from the set of all $P'$ ?
Any answer that can help me to have a better understanding of the subject is welcome