Deriving Optimal Skew Polycyclic Codes Over Fq Using Skew Polycyclic Linear Codes Over R = R1 ×R2 ×R3

Authors

  • Sassia Makhlouf Laboratory of Applications of Mathematics to Computer Science and Electronics, Faculty of Economic, Commercial, and Management Sciences, Batna 1 University, Batna, Algeria https://orcid.org/0000-0001-8479-9750
  • Karima Chatouh Laboratory of Applications of Mathematics to Computer Science and Electronics, Faculty of Economic, Commercial, and Management Sciences, Batna 1 University, Batna, Algeria https://orcid.org/0000-0003-4061-1239

DOI:

https://doi.org/10.37256/cm.5420245611

Keywords:

linear codes, skew polycyclic codes, dual codes, gray images, additive rings

Abstract

This paper investigates the theory and applications of linear and skew polycyclic codes over the ring mceclip0-4539bb3deae48df14ac741068c53b443.png = mceclip1-6fbe01b0d414845a0d95bfc189e057c0.png1 ×mceclip2.png2 ×mceclip3.png3, where mceclip4.pngi (0 i 3) are finite commutative rings. We first explore the structure of linear codes over mceclip6-29e8908958ed97e1a505a049978c8eb3.png, establishing foundational properties. Then, we introduce skew polycyclic codes over mceclip5-b127fd608b19e0d9cd7b5f181d310348.png, a generalization of polycyclic code over a finite field. We delve into the algebraic structure of these codes and demonstrate how they differ from their classical counterparts. Furthermore, we examine the dual codes ofskew polycyclic codes over mceclip7-838ad642efa6b8d2e89b7677cf1d872d.png, providing necessary and sufficient conditions for a code to be self-dual. Finally, we investigate the Gray images of skew polycyclic codes over mceclip7-838ad642efa6b8d2e89b7677cf1d872d.png, focusing on codes with optimal parameters. We provide explicit construction of Gray maps that yield images with good properties,such aslarge minimum distances and favorable automorphism groups. These results have potential applications in constructing new classes of error-correcting codes. We demonstrate this through an example of skew polycyclic codes applied in secret sharing schemes.

Downloads

Published

2024-11-29

How to Cite

1.
Makhlouf S, Chatouh K. Deriving Optimal Skew Polycyclic Codes Over F<i><sub>q</sub></i> Using Skew Polycyclic Linear Codes Over <i>R</i> = <i>R</i><sub>1</sub> ×<i>R</i><sub>2</sub> ×<i>R</i><sub>3</sub>. Contemp. Math. [Internet]. 2024 Nov. 29 [cited 2024 Dec. 4];5(4):5641-65. Available from: https://ojs.wiserpub.com/index.php/CM/article/view/5611