{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R244",
  "content_hash": null,
  "slug": "etc72-claim-open-status",
  "type": "claim",
  "title": "Existence remains open",
  "summary": "The audit found no accepted construction or nonexistence proof for an extremal Type II code of length 72.",
  "relevance": "For An extremal Type II binary code of length 72, record etc72-claim-open-status (“Existence remains open”) records a bound, answer, status fact, or structural consequence. The record states: The audit found no accepted construction or nonexistence proof for an extremal Type II code of length 72.",
  "relevance_source": "recorded",
  "body": "Sloane posed this exact question in 1973. A Type II code of length 72 has dimension 36, and the Mallows-Sloane bound gives minimum distance at most 16. The requested code would attain that bound.\n\nA 2025 primary preprint on small automorphism groups says that the best known minimum distance for binary self-dual length-72 codes is 12 and that the minimum-distance-16 question remains unsolved. Recent construction work continues to produce Type I and Type II \\([72,36,12]\\) codes. The primary-source audit through 2026-07-25 found no refereed construction of a \\([72,36,16]\\) code and no unrestricted exclusion.\n\nThe answer is recorded as open. The forced enumerator and automorphism results below are necessary conditions on a witness. Neither condition resolves the unrestricted case.",
  "status": "open",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "existence of a binary doubly-even self-dual code with parameters [72,36,16]",
    "bounds": {
      "length": {
        "min": 72,
        "max": 72
      },
      "dimension": {
        "min": 36,
        "max": 36
      },
      "minimum_distance": {
        "min": 16,
        "max": 16
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://www.preprints.org/manuscript/202506.2100",
      "locator": "On Small Automorphism Groups of Binary Self-Dual Codes, 2025, Section 6, which reports minimum distance 12 as the best known at length 72 and calls existence at distance 16 unsolved; origin checked against N. J. A. Sloane, IEEE Transactions on Information Theory 19(2) (1973), 251"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.preprints.org/manuscript/202506.2100",
    "locator": "On Small Automorphism Groups of Binary Self-Dual Codes, 2025, Section 6, which reports minimum distance 12 as the best known at length 72 and calls existence at distance 16 unsolved; origin checked against N. J. A. Sloane, IEEE Transactions on Information Theory 19(2) (1973), 251"
  },
  "models": [],
  "relations": [
    {
      "slug": "R243",
      "title": "Every witness has the same weight enumerator",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R242",
      "title": "Only five automorphism groups remain possible",
      "object_type": "claim",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R241",
      "title": "Construction and search audit",
      "object_type": "attempt",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "extremal-type-ii-code-72",
      "title": "extremal type ii code 72",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}