{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R243",
  "content_hash": null,
  "slug": "etc72-claim-forced-weight-enumerator",
  "type": "claim",
  "title": "Every witness has the same weight enumerator",
  "summary": "Gleason's theorem forces 249,849 words of weight 16 and fixes every other weight count.",
  "relevance": "For An extremal Type II binary code of length 72, record etc72-claim-forced-weight-enumerator (“Every witness has the same weight enumerator”) records a bound, answer, status fact, or structural consequence. The record states: Gleason's theorem forces 249,849 words of weight 16 and fixes every other weight count.",
  "relevance_source": "recorded",
  "body": "Set\n\\[\nA=x^8+14x^4y^4+y^8,\n\\qquad\nB=x^4y^4(x^4-y^4)^4.\n\\]\nGleason's theorem and the absence of weights 4, 8, and 12 force\n\\[\nW=A^9-126A^6B+3015A^3B^2-4398B^3.\n\\]\nExpanding gives\n\\[\n\\begin{aligned}\nW(1,y)={}&1+249849y^{16}+18106704y^{20}+462962955y^{24}\\\\\n&+4397342400y^{28}+16602715899y^{32}+25756721120y^{36}\\\\\n&+16602715899y^{40}+4397342400y^{44}+462962955y^{48}\\\\\n&+18106704y^{52}+249849y^{56}+y^{72}.\n\\end{aligned}\n\\]\nThe coefficients sum to \\(2^{36}\\), have the required complement symmetry, and are nonnegative. Sloane already listed these values in the 1973 problem statement.\n\nThe Assmus-Mattson theorem makes the supports at each nontrivial weight a 5-design. In particular, the 249,849 minimum-word supports form a\n\\[\n5\\text{-}(72,16,78)\n\\]\ndesign because\n\\[\n249849\\binom{16}{5}/\\binom{72}{5}=78.\n\\]\nThese arithmetic conditions are consistent and leave existence open.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "universal",
    "statement": "every binary doubly-even self-dual [72,36,16] code, if one exists"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1109/TIT.1973.1054975",
      "locator": "Sloane 1973, weight-distribution table and 5-(72,16,78) consequence; independently expanded and checked in etc72-artifact-enumerator-replay"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1109/TIT.1973.1054975",
    "locator": "Sloane 1973, weight-distribution table and 5-(72,16,78) consequence; independently expanded and checked in etc72-artifact-enumerator-replay"
  },
  "relations": [
    {
      "slug": "R240",
      "title": "Exact Gleason and 5-design parameter replay",
      "object_type": "artifact",
      "relation": "validates",
      "direction": "incoming"
    },
    {
      "slug": "R244",
      "title": "Existence remains open",
      "object_type": "claim",
      "relation": "informs",
      "direction": "outgoing"
    },
    {
      "slug": "extremal-type-ii-code-72",
      "title": "extremal type ii code 72",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}