{
  "abstract": "In this paper, we present three new constructions of optimal FDRM codes, all derived from subcodes of maximum rank-distance (MRD) codes. The first construction (Theorem~\\ref{theo5}) is based on a new family of generator matrices for systematic MRD codes and yields several previously unknown optimal FDRM codes. In particular, for $q\\geq 7$, it establishes the optimality of $[\\mathcal{F},7]_q$ FDRM codes with $ \\mathcal{F}=[1,2,3,4,8,8,8,8,8]$, thereby resolving an open problem posed by Zhang \\emph{et al.} (Des. Codes Cryptogr., 87(1):107--121, 2019). Our second construction exploits structural properties of generator matrices of a family of systematic MRD codes to obtain new optimal FDRM codes whenever each of the rightmost $δ-2$ columns of the Ferrers diagram $\\mathcal{F}$ contains at least $n-1$ dots. Building upon this approach, we further develop a third construction by substantially relaxing this requirement: it is sufficient to assume that each of the rightmost $δ-2$ columns of $\\mathcal{F}$ contains at least $n-r$ dots, where $r<κ$ and $κ=n-δ+1$. Furthermore, by exploiting the notion of proper combinations of Ferrers diagrams, we develop several recursive constructions that produce large FDRM codes from smaller building blocks, yielding a number of new optimal families. In particular, for an $n\\times n$ Ferrers diagram $\\mathcal{F}$ with prescribed parameters, one of these constructions establishes the optimality of $[\\mathcal{F},\\frac{n}{2}-1]_q$ FDRM codes whenever $n$ is even, thereby settling an open problem posed by Etzion \\emph{et al.} (IEEE Trans. Inf. Theory, 62(4):1616--1630, 2016).",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.24239",
  "author_total": 4,
  "authors": [
    {
      "name": "Fang-Wei Fu"
    },
    {
      "name": "Xuan Gao"
    },
    {
      "name": "Sihem Mesnager"
    },
    {
      "name": "Gang Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0tf35d6d9dwgj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.24239",
  "fulltext_url": null,
  "id": "0tf35d6d9dwgj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.24239",
  "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.IT"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7214011297",
  "pdf_url": "https://arxiv.org/pdf/2609.24239",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-21",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0tf35d6d9dwgj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:23:27Z",
      "native_id": "2609.24239",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.24239"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Optimal Ferrers Diagram Rank-Metric Codes: New Constructions, Diagram Combinations, and Applications to Constant-Dimension Subspace Codes",
  "title_text": "Optimal Ferrers Diagram Rank-Metric Codes: New Constructions, Diagram Combinations, and Applications to Constant-Dimension Subspace Codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:23:27Z",
  "url": "https://research.draconitic.com/paper/0tf35d6d9dwgj",
  "venue": "arXiv",
  "versions": []
}