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

Bush, Michael R.; Quijada, Danjoseph

doi:10.2140/involve.2021.14.361

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

Michael R. Bush and Danjoseph Quijada

Vol. 14 (2021), No. 3, 361–376

DOI: 10.2140/involve.2021.14.361

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 $𝔽_{q}$ 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 $𝔽_{q_{1}} \oplus \dots \oplus 𝔽_{q_{r}}$ .

Keywords

sequence, linear recurrence, period, characteristic polynomial, finite field, finite commutative ring, cyclic group algebra

Mathematical Subject Classification 2010

Primary: 11B50

Secondary: 11B37, 11B39, 94A55

Milestones

Received: 11 May 2018

Revised: 28 January 2021

Accepted: 19 February 2021

Published: 17 July 2021

Communicated by Filip Saidak

Authors

Michael R. Bush
Department of Mathematics Washington and Lee University Lexington, VA United States

Danjoseph Quijada
Department of Mathematics University of Southern California Los Angeles, CA United States

Vol. 14, No. 3, 2021