{
  "abstract": "This paper presents an algebraic method to construct convolutional codes with guaranteed \\emph{free and local minimum distances} without limit based on cyclic codes of odd lengths. The constructions are simple but effective, and no computer search is needed. For any two positive integers $r$ and $t$ with $1 \\leq r < t$, a rate-$r/t$ convolutional code $\\mathcal{C}_{\\text{convol}}$ can be constructed by using a chain of $r$ cyclic codes $\\mathcal{C}_0, \\mathcal{C}_1, \\ldots, \\mathcal{C}_{r-1}$ of the same length $n$ which satisfy the inclusion condition, $\\mathcal{C}_0 \\supset \\mathcal{C}_1 \\supset \\ldots \\supset \\mathcal{C}_{r-1}$. Such a convolutional code $\\mathcal{C}_{\\text{convol}}$ is composed of a \\emph{semi-infinite chain of identical local codes} confined in a diagonal band of width $n$. Each local code $\\mathcal{C}_{\\text{local}}$ of $\\mathcal{C}_{\\text{convol}}$ is formed from the $r$ cyclic codes in the code chain and is a specially localized subcode of the \\emph{mother code} $\\mathcal{C}_0$ in the code chain. The minimum distance $d_{\\text{local}}$ of each local code of $\\mathcal{C}_{\\text{convol}}$ is lower bounded by the minimum distance $d_0$ of the mother code $\\mathcal{C}_0$ in the code chain. The local structure of $\\mathcal{C}_{\\text{convol}}$ allows it to be decoded based on a designed parity-check matrix of the mother code $\\mathcal{C}_0$ using a sliding window decoding scheme.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.08296",
  "author_total": 4,
  "authors": [
    {
      "name": "Khaled Abdel-Ghaffar"
    },
    {
      "name": "Daniel J. Costello Jr."
    },
    {
      "name": "Juane Li"
    },
    {
      "name": "Shu Lin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eyt6hqjqr218p.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.08296",
  "fulltext_url": null,
  "id": "eyt6hqjqr218p",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.08296",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cs.IT"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7212037804",
  "pdf_url": "https://arxiv.org/pdf/2609.08296",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eyt6hqjqr218p.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2609.08296",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.08296"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Convolutional Codes from Cyclic Codes with Guaranteed Free and Local Minimum Distances",
  "title_text": "Convolutional Codes from Cyclic Codes with Guaranteed Free and Local Minimum Distances",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/eyt6hqjqr218p",
  "venue": "arXiv",
  "versions": []
}