{
  "abstract": "We extend the auxiliary-field method for scattered subspaces to evasive subspaces and subspace designs. For $1\\le h\\le k<hn$ and coprime $h,k+1$ with $n(k-h+1)\\ge k+1$, we construct spanning $(h,k)$-evasive $\\mathbb{F}_q$-subspaces of $\\mathbb{F}_{q^n}^r$ of dimension $rn(k-h+1)/(k+1)$ in every sufficiently large admissible ambient dimension, with a threshold independent of $q$. When $k<n$, this dimension is maximum. The proof combines an intersection lemma, a rank-support density inequality, scalar orbit counting, and a profile preserved by direct sums. The coding consequences include, for every fixed odd extension degree $m\\ge3$ and every sufficiently large $K$, minimal rank-metric codes of optimal length $2K$ with first two generalized weights $(m+1)/2$ and $m+1$. For these codes the stable profile is equivalent to minimality with the largest possible minimum distance; for even $m$ the analogous class consists of minimal MRD codes. More generally, for $1\\leτ\\le m-2$ and $\\gcd(m,τ+1)=1$, we obtain nondegenerate $τ$-minimal codes of length $(τ+1)K$ for every sufficiently large $K$, uniformly in $q$. We also construct maximum $h$-designs whose number of members grows exponentially with the ambient dimension at fixed $q,n,h$, and the associated maximum sum-rank distance (MSRD) codes with a growing number of equal long blocks. In extension degree five, the ordinary and $2$-minimal constructions hold for every $q$ and every $K\\ge32$ and $K\\ge59$, respectively.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2611.08627",
  "author_total": 1,
  "authors": [
    {
      "name": "Alessandro Giannoni"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/89kfkpa24d1jf.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2611.08627",
  "fulltext_url": null,
  "id": "89kfkpa24d1jf",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2611.08627",
  "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": null,
  "pdf_url": "https://arxiv.org/pdf/2611.08627",
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/89kfkpa24d1jf.ris",
  "sources": [
    {
      "first_seen": "2026-11-10T06:21:07Z",
      "native_id": "2611.08627",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2611.08627"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    }
  ],
  "title": "Evasive subspaces, minimal rank-metric codes and subspace designs via auxiliary fields",
  "title_text": "Evasive subspaces, minimal rank-metric codes and subspace designs via auxiliary fields",
  "type": "preprint",
  "updated_at": "2026-11-10T06:21:07Z",
  "url": "https://research.draconitic.com/paper/89kfkpa24d1jf",
  "venue": "arXiv",
  "versions": []
}