{
  "abstract": "There are eleven finite rings of order $p^{2}$ denoted by $A_p$ to $K_p$ in alphabetical order. In particular, we consider $I_{2}$ and $I_{3}$ which are commutative non-unital rings of orders 4 and 9 defined by generators and relations as \\[I_{p}=\\left\\langle a,b\\mid pa=pb=0,\\:a^{2}=b,\\:ab=0\\right\\rangle\\] for $p=2, 3,$ respectively. Alahmadi et al. studied codes over these rings. In this paper, we study additive complementary dual (ACD) codes over the rings $I_{2}$ and $I_{3}$. We show relations between ACD codes over $I_{2}$ and binary linear complementary dual (LCD) codes using a reduction map from $I_{2}$ to $\\mathbb{F}_{2}$, and between ACD codes over $I_{3}$ and ternary LCD codes using a reduction map from $I_{3}$ to $\\mathbb{F}_{3}$. Using the first relation, we classify ACD codes over $I_{2}$ with the highest minimum distances for $n=1, 2, 3$ and partially for $n=4, 5$. It turns out that they are maximum distance separable (MDS) codes. Using the second relation, we classify ACD codes over $I_{3}$ with the highest minimum Lee distances for $n=1, 2$ and partially for $n=3$. We generalize the two relations into a relation between ACD codes over $I_{p}$ and $p$-ary LCD codes using a reduction map from $I_{p}$ to $\\mathbb{F}_{p}$. This is a correction of the paper published in Advances in Mathematics of Communications, Volume 24, pages 61-76, 2026. In particular, we corrected the statements of Theorems 3.5, 4.7, 4.8, and their proofs.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Adv. Math. Commun. 24 (2026), 61-76",
  "arxiv": "2609.20190",
  "author_total": 3,
  "authors": [
    {
      "name": "Jon-Lark Kim"
    },
    {
      "name": "Marvin Olavides"
    },
    {
      "name": "Young Gun Roe"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/30hj378me9c87.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20190",
  "fulltext_url": null,
  "id": "30hj378me9c87",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20190",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.20190",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/30hj378me9c87.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20190",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20190"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Some MDS and ACD codes over commutative non-unital rings of orders 4 and 9 (Revision)",
  "title_text": "Some MDS and ACD codes over commutative non-unital rings of orders 4 and 9 (Revision)",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/30hj378me9c87",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.3934/amc.2026028",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}