On the k-order Jacobsthal polynomials and a new approach to coding theory

Yılmaz, Fatih; Arpacı, Samet; Ertaş, Aybüke; Rachidi, Mustapha

doi:10.47974/JDMSC-2398

Open Access Research Article

On the k-order Jacobsthal polynomials and a new approach to coding theory

^*Fatih YılmazCorresponding authorfatih.yilmaz@hbv.edu.trDepartment of MathematicsAnkara Hacı Bayram Veli UniversityAnkara, 06900, Turkey0000-0001-7873-1979View full profile → , Samet Arpacısamet.arpaci@hbv.edu.trDepartment of MathematicsAnkara Hacı Bayram Veli UniversityAnkara, 06900, Turkey0000-0002-6646-8197View full profile → , Aybüke Ertaşaybkerts99@gmail.comDepartment of MathematicsAnkara Hacı Bayram Veli UniversityAnkara, 06900, Turkey0000-0002-6133-5520View full profile → , Mustapha Rachidimustapha.rachidi@ufms.brIntitut of Mathematics, Universidade Federal de Mato Grosso do Sul Campo GrandeMato Grosso do Sul, MS- Brazil0000-0002-8210-7383View full profile →

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pp. 1707–1725Vol. 29Issue 4April 2026DOI: 10.47974/JDMSC-2398 Crossmark XML

Received:: 10 Dec 2024
Published Online:: 27 Dec 2025
Article type:: Research Article
Language:: EN
Article no.:: JDMSC-2398
Pages:: 1707–1725

Abstract

In this paper, we introduce a novel coding and decoding framework based on a generalized form of the k-order Jacobsthal polynomials. To this end, we first investigate the structure of these polynomials and derive a general expression for the powers of their associated companion matrices. Specifically, we develop a coding/decoding scheme utilizing integer powers of the companion matrix Qk (x) whose entries are determined by the coefficients of the k-order Jacobsthal polynomials. Furthermore, we define a suitable alphabet structured by generalized Jacobsthal numbers, integrating this alphabet into the proposed coding methodology. This construction enables a new perspective on algebraic coding theory by embedding the properties of k-order Jacobsthal polynomials into the design of the coding system. To demonstrate the practicality of the method, several illustrative examples and applications are presented. Finally, potential directions for future research and implications for coding theory are discussed.

Keywords

Jacobsthal polynomials k-order Jacobsthal polynomials Coding/decoding method

Subject Classifications

11B8311B3711T7194B35

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