{
  "abstract": "Linear complementary dual (LCD) codes are linear codes that intersect their dual trivially. We study double circulant (Euclidean) LCD codes over the finite chain ring $R=\\mathbb F_2+u\\mathbb F_2$, where $u^2=0$. Using the Chinese Remainder Theorem and the factorization of $x^n-1$, we derive an explicit enumeration formula, with a separate treatment of the constituent corresponding to $x-1$, which is not obtained by a formal specialization of the previously known counting formula over $\\mathbb F_q[u]/(u^k)$. Assuming Artin's primitive root conjecture for $2$, we show that the binary Gray images contain an asymptotically good family of rate $1/2$ whose relative distance can approach $H_2^{-1}(1/4)$ from below. More importantly from the constructive viewpoint, by combining Gray images with a known extension construction, we obtain a binary LCD code with parameters $[29,15,7]$ and construct two explicit permutation-inequivalent binary LCD codes with parameters $[45,23,9]$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.16121",
  "author_total": 3,
  "authors": [
    {
      "name": "Xingxing Xu"
    },
    {
      "name": "Minjia Shi"
    },
    {
      "name": "Jon-Lark Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/a09qzw1xv41cj.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.16121",
  "fulltext_url": null,
  "id": "a09qzw1xv41cj",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.16121",
  "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"
    ],
    [
      "arxiv",
      "math.RA"
    ],
    [
      "openalex",
      "1702|Coding theory and cryptography"
    ]
  ],
  "openalex": "W7221762661",
  "pdf_url": "https://arxiv.org/pdf/2610.16121",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-12",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/a09qzw1xv41cj.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:15Z",
      "native_id": "2610.16121",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.16121"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.RA",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Double circulant LCD codes over $\\mathbb{F}_2+u\\mathbb{F}_2$ and constructions for binary LCD codes",
  "title_text": "Double circulant LCD codes over 𝔽_2+u𝔽_2 and constructions for binary LCD codes",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:15Z",
  "url": "https://research.draconitic.com/paper/a09qzw1xv41cj",
  "venue": "arXiv",
  "versions": []
}