{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R241",
  "content_hash": null,
  "slug": "etc72-attempt-construction-and-search-audit",
  "type": "attempt",
  "title": "Construction and search audit",
  "summary": "Structured searches reach distance 12 or exclude selected symmetry classes; the unrestricted distance-16 question survives.",
  "relevance": "For An extremal Type II binary code of length 72, record etc72-attempt-construction-and-search-audit (“Construction and search audit”) documents a concrete method, search boundary, or failed route. The record states: Structured searches reach distance 12 or exclude selected symmetry classes; the unrestricted distance-16 question survives.",
  "relevance_source": "recorded",
  "body": "The audit separated constructions from symmetry exclusions. Yu, Zhu, and Wu use group rings and skew group rings to construct many binary self-dual codes of length 72. Their 2025 paper reports both singly-even and doubly-even codes with minimum distance 12. These are concrete near cases and supply no distance-16 generator matrix.\n\nFor the distance-16 target, several searches are exhaustive inside a stated automorphism class. Borello excludes automorphisms of order 6. Feulner and Nebe exclude groups containing \\(C_7\\), \\(C_3\\times C_3\\), or \\(D_{10}\\). Yorgov and Yorgov parameterize every possible order-4 case, reduce it by equivalence, and test the remaining branches. Their negative result excludes \\(C_4\\). The authors' supplement preserves the programs and run organization for that search.\n\nEach exclusion depends on the assumed symmetry. A code with trivial automorphism group remains among the five possible cases, so the union of these searches does not cover every binary Type II code of length 72.",
  "status": "completed",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "published construction and symmetry-restricted search results bearing on binary self-dual codes of length 72 through 2026-07-25",
    "bounds": {
      "length": {
        "min": 72,
        "max": 72
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "attempt",
    "citation": {
      "url": "https://doi.org/10.1109/TIT.2014.2313697",
      "locator": "Yorgov and Yorgov 2014, parameterization and exhaustive order-4 search; supplementary run materials at https://users.pfw.edu/yorgovd/code3/index.html; construction comparison from DOI 10.1007/s40314-024-03056-z"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1109/TIT.2014.2313697",
    "locator": "Yorgov and Yorgov 2014, parameterization and exhaustive order-4 search; supplementary run materials at https://users.pfw.edu/yorgovd/code3/index.html; construction comparison from DOI 10.1007/s40314-024-03056-z"
  },
  "relations": [
    {
      "slug": "R244",
      "title": "Existence remains open",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "extremal-type-ii-code-72",
      "title": "extremal type ii code 72",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}