{
  "abstract": "Let $\\mathbb L/\\mathbb K$ be a finite Abelian Galois extension with a fixed decomposition $\\mathrm{Gal}(\\mathbb L/\\mathbb K)=\\langleσ_1\\rangle\\times\\cdots\\times\\langleσ_m\\rangle$. We show that the quotient of the multivariate skew polynomial ring $\\mathbb L[x_1,\\ldots,x_m;σ_1,\\ldots,σ_m]$ by suitable central polynomials $H_1(x_1),\\ldots,H_m(x_m)$ is isomorphic to a product of $t$ copies of $\\mathrm{End}_{\\mathbb K}(\\mathbb L)$, so that one skew polynomial represents all blocks of a sum-rank metric space. The sum-rank weight of an element equals the rank of an associated Dickson matrix and of a generalized Moore matrix. Using this, we define sum-rank Reed--Muller-type codes and show that their dimension and minimum distance are given by the formulas for affine Cartesian codes with side lengths $\\mathrm{deg}\\,H_1,\\ldots,\\mathrm{deg}\\,H_m$; the same holds for codes supported on lower sets. The family contains as special cases linearized Reed--Solomon codes, Gabidulin codes, rank-metric Reed--Muller codes, and affine Cartesian codes. Over finite fields, every linearized Reed--Muller code of Berardini and Caruso is, after a monomial change of variables, a monomial code in our framework. This gives the dimension of all these codes, and their exact minimum distance in every degree whenever the support is a lower set.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.31358",
  "author_total": 2,
  "authors": [
    {
      "name": "Alessandro Neri"
    },
    {
      "name": "Giuseppe Zampella"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5hwqhawtafhva.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.31358",
  "fulltext_url": null,
  "id": "5hwqhawtafhva",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.31358",
  "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"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7223405813",
  "pdf_url": "https://arxiv.org/pdf/2610.31358",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5hwqhawtafhva.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.31358",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.31358"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Sum-rank analogues of Reed--Muller codes over Abelian Galois extensions",
  "title_text": "Sum-rank analogues of Reed--Muller codes over Abelian Galois extensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/5hwqhawtafhva",
  "venue": "arXiv",
  "versions": []
}