{
  "abstract": "This paper investigates cyclic codes over the commutative non-unital ring \\( \\mathbb{I} = \\langle a, B \\mid 2a = 2B = 0, a^2 = B, ab = 0 \\rangle \\), a ring of order four and characteristic two. We develop a structural description of linear and cyclic codes over \\( \\mathbb{I} \\), analyze their residue and torsion codes, and characterize self-dual codes. A Gray map from \\( \\mathbb{I}^n \\) to \\( \\mathbb{F}_{2^n} \\) is introduced and shown to transform cyclic codes into binary quasi-cyclic codes of index two. Numerical examples for lengths \\( 2 \\leq n \\leq 7 \\) illustrate the diversity of cyclic codes over \\( \\mathbb{I} \\) and highlight their potential applications.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": null,
  "author_total": 5,
  "authors": [
    {
      "name": "Adel Alahmadi"
    },
    {
      "name": "Tamador Alihia"
    },
    {
      "name": "Malak Altaiary"
    },
    {
      "name": "Abdul Nadim Khan"
    },
    {
      "name": "Patrick Solé"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eq4tpktq8ergv.bib",
  "cited_by": 0,
  "doi": "10.3390/e28111185",
  "fulltext_url": null,
  "id": "eq4tpktq8ergv",
  "issns": [
    "1099-4300"
  ],
  "landing_url": "https://hal.science/hal-05861490v1",
  "language": "en",
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "hal",
      "0.math"
    ],
    [
      "hal",
      "1.math.math-it"
    ],
    [
      "openalex",
      "2607|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223785463",
  "pdf_url": "https://hal.science/hal-05861490/document",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": "MDPI",
  "ris": "https://research.draconitic.com/paper/eq4tpktq8ergv.ris",
  "sources": [
    {
      "first_seen": "2026-11-13T12:20:03Z",
      "native_id": "hal-05861490",
      "source": "hal",
      "url": "https://hal.science/hal-05861490v1"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2607|Coding theory and cryptography",
      "subfield": "combinatorics"
    }
  ],
  "title": "Cyclic Codes over a Commutative Non-Unital Ring",
  "title_text": "Cyclic Codes over a Commutative Non-Unital Ring",
  "type": "peer-reviewed",
  "updated_at": "2026-11-13T12:20:03Z",
  "url": "https://research.draconitic.com/paper/eq4tpktq8ergv",
  "venue": "Entropy",
  "versions": []
}