{
  "abstract": "We prove that for every prime power $q$ and every integer $r\\geq 3$ there exists a maximum $2$-scattered $\\mathbb{F}_q$-subspace of $V(r,q^6)$, that is, a $2r$-dimensional $\\mathbb{F}_q$-subspace meeting every $2$-dimensional $\\mathbb{F}_{q^6}$-subspace in dimension at most $2$. Since an example in dimension three is known and direct sums preserve the property, it suffices to treat $r=4$ and $r=5$. For $r=4$ we study the subspaces $U_b=\\{(x,x^q+bx^{q^5}):\\mathrm{Tr}_{q^{12}/q^4}(x)=0\\}$ of $\\mathbb{F}_{q^{12}}^2$, $b\\in\\mathbb{F}_{q^{12}}$. We describe the parameters $b$ for which $U_b$ is not maximum $2$-scattered in terms of trace conditions and Dickson determinants, and deduce that at least $\\frac{1}{2}q^{12}-q^{11}+O(q^{10})$ parameters work; this number is positive for $q\\geq 3$, and a computer search settles $q=2$. For every $r\\geq 5$ coprime to $6$ we show that $U_s=\\{x+\\varepsilon s\\,x^{q^r}:x\\in\\mathbb{F}_{q^{2r}}\\}\\subseteq\\mathbb{F}_{q^{6r}}$ is maximum $2$-scattered for all $\\varepsilon\\in\\mathbb{F}_{q^6}\\setminus\\mathbb{F}_{q^2}$ and $s\\in\\mathbb{F}_{q^r}\\setminus\\mathbb{F}_q$. Consequently, $\\mathbb{F}_{q^6}$-linear $[2r,r,4]_{q^6/q}$ MRD codes exist for every $r\\geq 3$; for $r$ coprime to $6$, the codes obtained from $U_s$ are equivalent to their duals.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.08237",
  "author_total": 5,
  "authors": [
    {
      "name": "Daniele Bartoli"
    },
    {
      "name": "Francesco Ghiandoni"
    },
    {
      "name": "Alessandro Giannoni"
    },
    {
      "name": "Giuseppe Marino"
    },
    {
      "name": "Alessandro Neri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8a8kgjh69bv12.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.08237",
  "fulltext_url": null,
  "id": "8a8kgjh69bv12",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.08237",
  "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",
      "math.CO"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7221061332",
  "pdf_url": "https://arxiv.org/pdf/2610.08237",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-14",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8a8kgjh69bv12.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:35Z",
      "native_id": "2610.08237",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.08237"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Maximum 2-scattered subspaces of $V(r,q^6)$ for every $r\\geq 3$",
  "title_text": "Maximum 2-scattered subspaces of V(r,q^6) for every r≥ 3",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:35Z",
  "url": "https://research.draconitic.com/paper/8a8kgjh69bv12",
  "venue": "arXiv",
  "versions": []
}