Cyclic Codes over a Commutative Non-Unital Ring
This paper investigates cyclic codes over the commutative non-unital ring I=⟨a,b∣2a=2b=0,a2=b,ab=0⟩, a ring of order four and characteristic two. We develop a structural description of linear and cyclic codes over I, analyze their residue and torsion codes, and characterize self-dual codes. A Gray map from In to F22n is introduced and shown to transform cyclic codes into binary quasi-cyclic codes of index two. Numerical examples for lengths 2≤n≤7 illustrate the diversity of cyclic codes over I and highlight their potential applications.
Authors
- Abdul Nadim Khan (ORCID: https://orcid.org/0000-0002-5861-6137)
- Patrick Solé (ORCID: https://orcid.org/0000-0002-4078-8301)
- Adel N. Alahmadi (ORCID: https://orcid.org/0000-0002-7758-3537)
- Tamador Alihia
- M. M. Altaiary (ORCID: https://orcid.org/0000-0003-2621-0498)
Institutions
- Centre National de la Recherche Scientifique (FR)
- Qassim University (SA)
- King Abdulaziz University (SA)
- Shaqra University (SA)
- Aix-Marseille Université (FR)
- Institut de Mathématiques de Marseille (FR)
- Centrale Méditerranée (FR)
Publication Details
- Journal
- Entropy
- Published
- 2026-10-27
- DOI
- https://doi.org/10.3390/e28111185
- Primary Topic
- Coding theory and cryptography
- Type
- article
- Field-Weighted Citation Impact
- 0.00
Funders
- King Abdulaziz University