{
  "abstract": "Let $1\\le h\\le n-2$ and $g=\\gcd(n,h+1)$. We prove that, for every prime power $q$, a maximum $h$-scattered $\\mathbb{F}_q$-subspace of $V(r,q^n)$ exists whenever $(h+1)\\mid rn$ and $$r\\ge\\frac{h+1}{g}\\bigl(h(n-h-2)+1\\bigr).$$ In particular, for fixed $n$ and minimum distance $δ\\in\\{2,\\ldots,n-1\\}$, $\\mathbb{F}_{q^n}$-linear MRD codes of length $N$ exist for every $q$ and every sufficiently large $N$ such that $n$ divides $(n-δ+1)N$, with a bound on $N$ depending only on $n$ and $δ$. We take the $\\mathbb{F}_{q^s}$-span of the columns of a random matrix $A$ over the algebra $R=\\mathbb{F}_{q^n}\\otimes_{\\mathbb{F}_q}\\mathbb{F}_{q^s}$, for a suitable $s$. Such a subspace is maximum $h$-scattered if and only if the kernel of $A$ contains no nonzero vector whose expansion with respect to an $\\mathbb{F}_q$-basis of $\\mathbb{F}_{q^s}$ has rank weight at most $h+1$. When $\\gcd(n,s)=1$, $R$ is a field, and a first-moment argument, counting these vectors up to multiplication by $\\mathbb{F}_{q^n}^*\\mathbb{F}_{q^s}^*$, shows that with positive probability the kernel contains none of them if $s>h(n-h-2)$. When $d=\\gcd(n,s)>1$, $R$ is a product of $d$ fields, and vectors vanishing in some factors lie in the kernel with larger probability. We control them by proving that, for an $(h+1)$-dimensional $\\mathbb{F}_q$-subspace $D$ of $\\mathbb{F}_{q^n}$, the elements of $D\\otimes_{\\mathbb{F}_q}\\mathbb{F}_{q^s}$ vanishing outside $k$ given factors, $0<k<d$, form an $\\mathbb{F}_{q^s}$-space of dimension at most $(h+1)k/d$, with equality only if $D$ is linear over a subfield of $\\mathbb{F}_{q^n}$ larger than $\\mathbb{F}_q$. Taking $s=gr/(h+1)$ gives the bound on $r$. Finally, if $(n-h-1)\\mid(h+1)$, then for every $r$ satisfying the above inequality there is an $h$-scattered $\\mathbb{F}_q$-subspace of $V(r,q^n)$ of dimension $\\lfloor rn/(h+1)\\rfloor$, without a divisibility condition on $rn$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.32737",
  "author_total": 2,
  "authors": [
    {
      "name": "Alessandro Giannoni"
    },
    {
      "name": "Alessandro Neri"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bankd0pf72sb9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.32737",
  "fulltext_url": null,
  "id": "bankd0pf72sb9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.32737",
  "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": "W7223663856",
  "pdf_url": "https://arxiv.org/pdf/2610.32737",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-23",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bankd0pf72sb9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:19Z",
      "native_id": "2610.32737",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.32737"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "A probabilistic construction of maximum $h$-scattered subspaces",
  "title_text": "A probabilistic construction of maximum h-scattered subspaces",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:19Z",
  "url": "https://research.draconitic.com/paper/bankd0pf72sb9",
  "venue": "arXiv",
  "versions": []
}