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  "slug": "modular-code-obstructions",
  "title": "When algebraic symmetry weakens a code",
  "shortTitle": "Modular-code distance obstructions",
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  "oneLine": "A short-support mechanism limits error protection in a specified family of symmetric codes—even when their information rate stays positive.",
  "abstract": "A written proof candidate shows that a characteristic-two XOR-circulant graph code of matrix order n=2^r has a subcode of dimension at least n/4 supported on at most n coordinates whenever its coefficient span has binary dimension at most r. It also proves an all-prime vector-valued quotient-concentration theorem, yielding relative distance O(1/r) for free modular group-algebra codes at fixed field and index as elementary-abelian rank r grows. The revision gives a sharp characteristic-two alphabet threshold using a classical Cauchy construction and explicit boundary examples. Scalar binary H-code badness is credited to Charpin; the finite attaining formula is Li–Wang.",
  "datePublished": "2026-09-29",
  "dateModified": "2026-09-29",
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  "doi": "10.5281/zenodo.23046770",
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  "problem": {
    "name": "How does modular algebraic structure limit code distance?",
    "url": "https://www.numdam.org/item/BURO_1983__41__3_0/"
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  "keywords": [
    "coding theory",
    "minimum distance",
    "modular group algebras",
    "H-codes",
    "Hadamard matrices",
    "quotient concentration"
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    "Coefficient-span e≤r forces a subcode of dimension at least n/4 on at most n of 2n coordinates.",
    "For n≥8, the minimum characteristic-two alphabet admitting Hadamard branch number at least n is 2n; a classical Cauchy matrix attains n+1.",
    "Vector-valued quotient concentration holds at an explicit sufficient threshold in every prime characteristic.",
    "At fixed field, index and positive free rank, relative distance is O(1/r) as elementary-abelian rank grows."
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  "evidencePackage": "Written analytic proofs; 2,097,152-kernel exact checker, 30 concentration certificates, finite minor controls and semantic rejection tests.",
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  "relatedWorks": [
    {
      "citation": "Charpin (1983), Étude sur la valuation des H-codes binaires. The binary scalar bound and asymptotic badness are predecessors, not new results here.",
      "url": "https://www.numdam.org/item/BURO_1983__41__3_0/",
      "doi": null
    },
    {
      "citation": "Charpin (1996), Self-dual codes which are principal ideals of the group algebra. Later treatment of the binary H-code mechanism.",
      "url": "https://doi.org/10.1016/0378-3758(95)00011-9",
      "doi": "10.1016/0378-3758(95)00011-9"
    },
    {
      "citation": "Li and Wang (2017), Design of lightweight linear diffusion layers from near-MDS matrices. Their formula supplies the finite attaining example.",
      "url": "https://doi.org/10.13154/tosc.v2017.i1.129-155",
      "doi": "10.13154/tosc.v2017.i1.129-155"
    },
    {
      "citation": "Borello and Willems, On the algebraic structure of quasi group codes. Free coordinate actions and module structure must be distinguished.",
      "url": "https://arxiv.org/abs/1912.09167",
      "doi": "10.48550/arXiv.1912.09167"
    },
    {
      "citation": "Fan and Lin (2022), Asymptotic properties of quasi-group codes. Good families in other limits do not contradict this restricted modular obstruction.",
      "url": "https://arxiv.org/abs/2203.00958",
      "doi": "10.48550/arXiv.2203.00958"
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