Period sets of linear recurrences over finite fields and related commutative rings

Abstract

After giving an overview of the existing theory regarding the periods of sequences defined by linear recurrences over finite fields, we give explicit descriptions of the sets of periods that arise if one considers all sequences over ${\mathbb{𝔽}}_q$ generated by linear recurrences for a fixed choice of the degree $k$ in the range $1\le k\le 4$. We also investigate the periods of sequences generated by linear recurrences over rings of the form ${\mathbb{𝔽}}_{q_1}\oplus \cdots \oplus {\mathbb{𝔽}}_{q_r}$.