Let $p$ be a prime number with $p\neq 2$. We consider sequences generated by $n$th order linear recurrence relations over the finite field $Z_p$. In the first part of this paper we generalize some of the ideas in  to $n$th order linear recurrences. We then consider the case where the characteristic polynomial of the recurrence has one root in $Z_p$ of multiplicity $n$. In this case, we show that the corresponding recurrence can be generated by a relatively simple matrix.
Thomas McKenzie. Shannon Overbay. Robert Ray. "Jordan Forms and $n$th Order Linear Recurrences." Missouri J. Math. Sci. 26 (2) 122 - 133, November 2014. https://doi.org/10.35834/mjms/1418931954