{
  "abstract": "For a matrix rank-metric code $\\mathcal{C}$ and an integer $i$, the $i$-th subcode distance of $\\mathcal{C}$ is the largest minimum rank distance of an $i$-dimensional subcode of $\\mathcal{C}$. We study how subcode distances change under the Add-and-Remove construction, in which a subcode of codimension $\\ell_s$ is kept and a space of dimension $\\ell_a$ is added; this construction is used, with Gabidulin codes, in the $\\mathsf{MIRANDA}$ signature scheme. For arbitrary matrix codes, we give inequalities between the subcode distances of the original and of the new code, and we show that Delsarte duality interchanges $\\ell_s$ and $\\ell_a$. For matrix Gabidulin codes, whose subcode distances are given by an explicit formula, we show that, if $\\ell_s<\\max\\{m,n\\}$, then within each block of indices $(a-1)\\max\\{m,n\\}+1,\\ldots,a\\max\\{m,n\\}$ not exceeding $\\dim_{\\mathbb{F}_q}(\\mathcal{C}_s)$, all but the last $\\ell_s$ subcode distances are the same as for the Gabidulin code, and the remaining ones decrease by at most one; a small example shows that such a decrease can occur. We also express the minimum distance of the new code in terms of the rank distance between the added space and the common subcode. For $\\mathsf{MIRANDA}$, the lower bound on the minimum distance of the dual public code used by the known distinguisher when $\\ell_s=0$ is a special case of these results; for the proposed parameters, where $\\ell_s>0$, the same bound holds for a subcode of codimension $\\ell_s$ of the dual public code.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.03460",
  "author_total": 3,
  "authors": [
    {
      "name": "Gianira N. Alfarano"
    },
    {
      "name": "Rakhi Pratihar"
    },
    {
      "name": "Adrien Vinçotte"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/6p4bxt2fd7qm2.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.03460",
  "fulltext_url": null,
  "id": "6p4bxt2fd7qm2",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.03460",
  "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.IT"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.03460",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/6p4bxt2fd7qm2.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2610.03460",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.03460"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Code distances of matrix rank-metric codes under Add-and-Remove transformations",
  "title_text": "Code distances of matrix rank-metric codes under Add-and-Remove transformations",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/6p4bxt2fd7qm2",
  "venue": "arXiv",
  "versions": []
}