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

TL;DR

This paper characterizes the sets of possible periods for sequences generated by linear recurrences over finite fields and certain rings, providing explicit descriptions for degrees up to 4 and exploring more complex ring structures.

Contribution

It offers explicit descriptions of period sets for linear recurrences over finite fields of degree up to 4 and extends the analysis to sequences over composite rings.

Findings

Explicit period sets for degrees 1 to 4 over finite fields

Descriptions of periods over direct sums of finite fields

Analysis of sequences over rings of the form _{q_1} \u2297 _{q_r}

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 $1 \leq k \leq 4$. We also investigate the periods of sequences generated by linear recurrences over rings of the form $\mathbb{F}_{q_1} \oplus \ldots \oplus \mathbb{F}_{q_r}$.