{
  "abstract": "We study non-additive rank-metric codes contained in the space of symmetric matrices over a finite field. Our main focus is on symmetric $2$-codes. For odd $q$, we introduce a quotient construction for $\\mathrm{Sym}_3(q)$ in which rank-one differences are described by isotropic vectors of a nondegenerate quadratic form on $\\mathbb F_q^4$. This leads to a correspondence with partial ovoids of the elliptic quadric $Q^-(5,q)$. Using a construction of Cossidente and Pavese, we obtain a non-additive symmetric $2$-code in $\\mathrm{Sym}_3(q)$ of cardinality $q^4+2q^3$, improving the previously known general lower bound. We also give an explicit matrix description of this family. Lastly, we present a block-based construction for symmetric $2$-codes that extends the three-dimensional improvement to all odd dimensions. In particular, for every odd $n\\geq3$ and every prime power $q>2$, we obtain \\begin{equation*} A_q^{\\rm sym}(n,2) \\ge q^{(n^2-1)/2}+q^{(n^2-3)/2}+q^{(n^2-9)/2}, \\end{equation*} which is strictly larger than the maximum cardinality of an additive symmetric $2$-code with the same parameters. This yields the first infinite family (as the matrix dimension $n$ varies) of non-additive symmetric $2$-codes whose size is larger than the largest achievable size of an additive code with the same parameters; before this, such a gap had only been established for $n=3$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18061",
  "author_total": 2,
  "authors": [
    {
      "name": "Marta Messia"
    },
    {
      "name": "Ferdinando Zullo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bna016swdex3q.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18061",
  "fulltext_url": null,
  "id": "bna016swdex3q",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18061",
  "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.CO"
    ],
    [
      "arxiv",
      "math.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7221777551",
  "pdf_url": "https://arxiv.org/pdf/2610.18061",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-13",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bna016swdex3q.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:19Z",
      "native_id": "2610.18061",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18061"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Non-additive symmetric rank-metric codes",
  "title_text": "Non-additive symmetric rank-metric codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:19Z",
  "url": "https://research.draconitic.com/paper/bna016swdex3q",
  "venue": "arXiv",
  "versions": []
}