{
  "abstract": "It is well known that the length $n$ of a binary Type II code is divisible by $8$. The classical proof relies on Gleason's theorem on weight enumerators. Motivated by a question of Vera Pless, we give elementary proofs of this fact that avoid invariant theory. Our main tool is a congruence modulo $8$ for the determinant of the integer Gram matrix $AA^{\\mathsf T}$ of a binary orthogonal matrix $A$, obtained directly from the Leibniz expansion. It gives a short proof of a theorem of Janusz: the number of rows of weight congruent to $3 \\pmod 4$ in a binary orthogonal matrix is divisible by $4$. From this single statement we derive both the length restriction for Type II codes and the theorem of Conway and Sloane that every vector in the shadow of a binary self-dual code of length $n$ has weight congruent to $n/2 \\pmod 4$. We also give two short character-theoretic proofs of the length restriction and explain how all of these arguments detect the Arf invariant of a quadratic form.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.30809",
  "author_total": 2,
  "authors": [
    {
      "name": "Jae-Hyun Baek"
    },
    {
      "name": "Jon-Lark Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/bf5x4ym7rng3c.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.30809",
  "fulltext_url": null,
  "id": "bf5x4ym7rng3c",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.30809",
  "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": "W7223282308",
  "pdf_url": "https://arxiv.org/pdf/2610.30809",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/bf5x4ym7rng3c.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:38Z",
      "native_id": "2610.30809",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.30809"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.IT",
      "subfield": "general"
    }
  ],
  "title": "An integer determinant approach to the length of Type II codes and the weights of shadows",
  "title_text": "An integer determinant approach to the length of Type II codes and the weights of shadows",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/bf5x4ym7rng3c",
  "venue": "arXiv",
  "versions": []
}