{
  "abstract": "Let $q=3^m$, let $K=\\mathbb F_{q^2}$, and let \\[\\mathcal T=\\{x\\in K^*:\\operatorname{N}_{K/\\mathbb F_q}(x)=1\\}.\\] For both cyclic and constacyclic Gashkov-Sidel'nikov codes, we show that the set of signed parity-check column labels is precisely $\\mathcal T$. Consequently, the decoding problem separates into two stages: determining the minimum error weight associated with a syndrome $S$ and constructing an error vector attaining this minimum. We identify the former quantity with the minimum additive length of $S$ with respect to $\\mathcal T$ and determine it exactly by the norm and the quadratic character of $\\mathbb F_q$. We also determine the complete coset-weight distribution and recover the known covering radius $3$. For the constructive part, we use quadratic-character sums and Weil bounds to construct a coset leader for every syndrome of coset weight three. The resulting procedures give complete maximum-likelihood decoders.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.20402",
  "author_total": 5,
  "authors": [
    {
      "name": "Minjia Shi"
    },
    {
      "name": "Shitao Li"
    },
    {
      "name": "Yuhong Xia"
    },
    {
      "name": "Tor Helleseth"
    },
    {
      "name": "Ferruh Ozbudak"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eb2tsasw6j0y3.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.20402",
  "fulltext_url": null,
  "id": "eb2tsasw6j0y3",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.20402",
  "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.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7213603686",
  "pdf_url": "https://arxiv.org/pdf/2609.20402",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-17",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eb2tsasw6j0y3.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:22:23Z",
      "native_id": "2609.20402",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.20402"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Norm-One Torus Decompositions and Decoding of Gashkov-Sidel'nikov Codes",
  "title_text": "Norm-One Torus Decompositions and Decoding of Gashkov-Sidel'nikov Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:22:23Z",
  "url": "https://research.draconitic.com/paper/eb2tsasw6j0y3",
  "venue": "arXiv",
  "versions": []
}