{
  "abstract": "Given a cyclic code $C_g = \\langle g(x) \\rangle$ of odd length $m$ over $\\mathbb{Z}_4$, one common way to build a quasi-cyclic (QC) code is to pick $f_1, \\dots, f_\\ell \\in \\mathbb{Z}_4[x]$ and let $C = \\langle (f_1 g, \\dots, f_\\ell g) \\rangle$. Because $\\mathbb{Z}_4$ is not a field, the type of $C_g$ ($4^{k_1}2^{k_2}$) is not necessarily inherited by $C$. Determining conditions under which the type of $C_g$ is inherited by $C$ was posed as an open problem recently in \\cite{AydinLuOnta2023}. In this paper, we settle this problem. We first give two sufficient conditions for the type to be preserved: one requires a single $f_i$ to be coprime to $x^m-1$ over $\\mathbb{F}_2$, the other only requires the $f_i$ to be jointly coprime to it. Neither condition is necessary in general. Using the fact that $x^m-1$ is squarefree for odd $m$, we decompose $\\mathbb{Z}_4[x]/\\langle x^m-1\\rangle$ into a product of finite chain rings using Chinese remainder theorem (CRT) and derive a condition on $f_1, \\dots, f_\\ell$ that is both necessary and sufficient for $C$ to match the type of $C_g$. This condition depends only on the irreducible factors of $x^m-1$ where $g$ does not already vanish. This also yields a simple test for when $C$ is a free $\\mathbb{Z}_4$-module. Finally, we report many new QC codes over $\\mathbb{Z}_4$, found by computer searches using Magma software~\\cite{Magma1997} guided by this criterion, with Lee distances greater than previously known codes of the same type.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.21601",
  "author_total": 2,
  "authors": [
    {
      "name": "Nuh Aydin"
    },
    {
      "name": "Aditya Tyagi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4ch9v811sfzsr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.21601",
  "fulltext_url": null,
  "id": "4ch9v811sfzsr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.21601",
  "language": null,
  "licence": "CC-BY-NC-SA-4.0",
  "licence_label": "CC BY-NC-SA 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by-nc-sa/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2609.21601",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-18",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4ch9v811sfzsr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:07Z",
      "native_id": "2609.21601",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.21601"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Resolution of an Open Problem on Quasi-Cyclic Codes over $\\mathbb{Z}_4$ and New Quaternary Linear Codes",
  "title_text": "Resolution of an Open Problem on Quasi-Cyclic Codes over ℤ_4 and New Quaternary Linear Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:07Z",
  "url": "https://research.draconitic.com/paper/4ch9v811sfzsr",
  "venue": "arXiv",
  "versions": []
}