{
  "abstract": "Given a linear code C over Fq with generator matrix G=[X∣Z], we study the nonstandard two-coordinate symplectic extension G¯(u,v)=[X∣Xu∣Z∣Zv] and its effects on the symplectic hull and minimum symplectic distance. The extension transforms the symplectic Gram matrix according to M¯=M+pf⊤−fp⊤, where p=Xu and f=Zv, and hence induces an alternating perturbation of rank at most two. We characterize all possible hull transitions, showing that the symplectic hull dimension can change only by −2, 0, or +2, and derive necessary and sufficient conditions for the corresponding cases. These results yield explicit criteria for symplectic LCD preservation, one-step hull reduction, a mother-code criterion for complete iterated hull reduction, and minimum-distance improvement. We further obtain exact counting formulas for admissible extension pairs, including inclusion–exclusion formulas under simultaneous structural and metric constraints, together with an automorphism-orbit reduction for eliminating equivalent extensions. In the binary case, the standard symplectic–quaternary correspondence transfers the framework to one-symbol extensions of quaternary trace-Hermitian additive complementary dual codes and yields a conditional comparison criterion with the Hermitian LCD subclass. Exhaustive computations over F2 and F3 provide finite-instance validation of the predicted hull transitions, distance behavior, and exact counts.",
  "abstract_shown": true,
  "affiliations": [
    "College of Computer Engineering, Jimei University, Xiamen 361021, China",
    "School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin Institute of Technology, Harbin 150001, China"
  ],
  "article_type": null,
  "arxiv": null,
  "author_total": 4,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Kangda Cheng"
    },
    {
      "affs": [
        0
      ],
      "name": "Weilin Wu",
      "orcid": "0009-0004-0524-1951"
    },
    {
      "affs": [
        1
      ],
      "name": "Xiaobao Li",
      "orcid": "0009-0001-5416-3943"
    },
    {
      "affs": [
        1
      ],
      "name": "Jinlong Liu",
      "orcid": "0000-0002-0284-5029"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/118crrwdzpzd0.bib",
  "cited_by": 0,
  "doi": "10.3390/math14214063",
  "fulltext_url": null,
  "id": "118crrwdzpzd0",
  "issns": [
    "2227-7390"
  ],
  "landing_url": "https://www.mdpi.com/2227-7390/14/21/4063",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [],
  "openalex": null,
  "pdf_url": null,
  "pmcid": null,
  "pmid": null,
  "published": "2026-11-05",
  "publisher": "MDPI AG",
  "ris": "https://research.draconitic.com/paper/118crrwdzpzd0.ris",
  "sources": [
    {
      "first_seen": "2026-11-06T00:21:06Z",
      "native_id": "10.3390/math14214063",
      "source": "crossref",
      "url": "https://www.mdpi.com/2227-7390/14/21/4063"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "issn:2227-7390",
      "subfield": "general"
    }
  ],
  "title": "Symplectic Hull Control via Nonstandard Two-Coordinate Extensions and Trace-Hermitian ACD Codes",
  "title_text": "Symplectic Hull Control via Nonstandard Two-Coordinate Extensions and Trace-Hermitian ACD Codes",
  "type": "peer-reviewed",
  "updated_at": "2026-11-07T00:21:04Z",
  "url": "https://research.draconitic.com/paper/118crrwdzpzd0",
  "venue": "Mathematics",
  "versions": []
}