{
  "abstract": "We study $\\mathbb F_q$-linear sets whose positive point weights belong to a prescribed finite set $A$. The integer $e=\\gcd(A)$ always divides the rank of the defining subspace, but the divisibility does not by itself imply linearity over $\\mathbb F_{q^e}$. We show that it does in a broad parameter range. Let $L_U$ be an $\\mathbb F_q$-linear set of rank $N$ spanning $PG(h-1,q^m)$, $h\\ge2$, without being equal to it, and assume that $q\\ge m$ and that the minimum weight of its points is $a\\ge2$. Combining the theorem of Csajbók, Marino and Pepe on the maximum field of linearity with the lower bound of Adriaensen and Santonastaso for the size of a linear set containing a point of weight one, we prove that $L_U$ is also defined by an $\\mathbb F_{q^t}$-subspace, for some divisor $t$ of $m$, whose $\\mathbb F_q$-dimension is exactly $N+t-a$. Under a natural gap condition excluding divisors of $m$ in the interval $(e,N-e]$, this forces $a=e$ and the defining subspace to be $\\mathbb F_{q^e}$-linear. Through geometric duality, the point weights of the dual of a system are the deficiencies $m-w$ of the rank weights $w$ of the associated code. For rank-metric codes in which the subcode generated by the codewords of nonmaximum rank contains codewords of maximum rank, we show that deleting suitable coordinates yields the $\\mathbb F_q$-expansion of a code over $\\mathbb F_{q^t}$. We show that the codes in which the codewords of nonmaximum rank generate a subcode without codewords of maximum rank are exactly the direct sums of a simplex code and a code of length smaller than $m$. We derive field-reduction results for two-weight and three-weight rank-metric codes, and show that no gap condition is needed when $m$ divides the length or when the code has only two nonzero weights. Finally, we study two-dimensional codes of length at most $m$ and provide families of three-weight examples.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08044",
  "author_total": 4,
  "authors": [
    {
      "name": "Jan De Beule"
    },
    {
      "name": "Leen Demuys"
    },
    {
      "name": "Ferdinando Zullo"
    },
    {
      "name": "Olga Polverino"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/55b1gw6pmt96w.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08044",
  "fulltext_url": null,
  "id": "55b1gw6pmt96w",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08044",
  "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",
      "math.CO"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08044",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/55b1gw6pmt96w.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T18:21:34Z",
      "native_id": "2611.08044",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08044"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Few-Weight Linear Sets and Divisibility of Rank-Metric Codes",
  "title_text": "Few-Weight Linear Sets and Divisibility of Rank-Metric Codes",
  "type": "preprint",
  "updated_at": "2026-11-09T00:20:48Z",
  "url": "https://research.draconitic.com/paper/55b1gw6pmt96w",
  "venue": "arXiv",
  "versions": []
}