I have a $n\times n$ matrix with coefficients from a finite field $F$. For any vector $v$ of $F^n$, I consider the sequence:
$v_1=v$ and $v_{n+1}=M.v_n$
The rank of $v$ is the rank of the sequence $v_i$ : it is the maximum integer $k$, such that $v_1,v_2,\dots,v_k$ are linearly independent.
How to find a vector $v$ of maximal rank ?
What I tried : I know there is some minimal polynomial $P\in F[x]$ such that $P(M)=0$. Hence the maximal rank is the degree of $P$. But how to find $v$ efficiently (I know that I can try all vectors of $F^n$, as $F^n$ is finite, but I would like to find some more efficient algorithm to find the vector).