{
  "abstract": "This paper determines the exact lengths of shortest self-orthogonal and LCD embeddings of linear codes over $\\mathbb{F}_q+u\\mathbb{F}_q$. By decomposing Gram matrices over $\\mathbb{F}_q+u\\mathbb{F}_q$ into pairs of symmetric matrices over the finite field $\\mathbb{F}_q$, the embedding problems are reduced to the congruence classification of symmetric and alternate matrices over finite fields. Complete formulas for the shortest self-orthogonal embedding length are obtained, with two distinct cases arising in both even and odd characteristic. We also show that every self-orthogonal code over $\\mathbb{F}_q+u\\mathbb{F}_q$ with nonzero free rank can be viewed as a shortest self-orthogonal embedding of another code. We use Witt theory to construct all shortest self-orthogonal embeddings. A complete characterization of shortest LCD embeddings is also established in terms of invertible and arbitrary matrices of prescribed sizes appended to a generator matrix. Examples of self-orthogonal and LCD embeddings with the largest minimum distance for the code considered are also presented, some of whose Gray images are optimal codes over $\\mathbb{F}_q$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28222",
  "author_total": 2,
  "authors": [
    {
      "name": "Junmin An"
    },
    {
      "name": "Jon-Lark Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7xbx3ay6wyfgr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28222",
  "fulltext_url": null,
  "id": "7xbx3ay6wyfgr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28222",
  "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",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7204825291",
  "pdf_url": "https://arxiv.org/pdf/2608.28222",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7xbx3ay6wyfgr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:28Z",
      "native_id": "2608.28222",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28222"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "Shortest self-orthogonal and LCD embeddings of linear codes over Fq+uFq",
  "title_text": "Shortest self-orthogonal and LCD embeddings of linear codes over Fq+uFq",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:28Z",
  "url": "https://research.draconitic.com/paper/7xbx3ay6wyfgr",
  "venue": "arXiv",
  "versions": []
}