{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R242",
  "content_hash": null,
  "slug": "etc72-claim-automorphism-restrictions",
  "type": "claim",
  "title": "Only five automorphism groups remain possible",
  "summary": "A witness has automorphism group C1, C2, C3, C2 by C2, or C5.",
  "relevance": "For An extremal Type II binary code of length 72, record etc72-claim-automorphism-restrictions (“Only five automorphism groups remain possible”) records a bound, answer, status fact, or structural consequence. The record states: A witness has automorphism group C1, C2, C3, C2 by C2, or C5.",
  "relevance_source": "recorded",
  "body": "The published exclusions reduce the full permutation automorphism group to\n\\[\nC_1,\\quad C_2,\\quad C_3,\\quad C_2\\times C_2,\\quad\\text{or}\\quad C_5.\n\\]\nBorello's 2014 theorem first reduces the list to cyclic groups of orders 1 through 5 and the elementary abelian group of order 4. Yorgov and Yorgov then exclude an element of order 4 by an exhaustive parameterized search, removing \\(C_4\\).\n\nThe restrictions also determine the possible cycle structures. Every involution is fixed-point-free, so it acts as 36 transpositions. Every element of order 3 is fixed-point-free, so it acts as 24 three-cycles. An element of order 5 acts as fourteen 5-cycles and fixes two coordinates.\n\nEarlier primary computations exclude \\(C_7\\), \\(C_3\\times C_3\\), \\(D_{10}\\), elements of order 6, \\(S_3\\), \\(A_4\\), \\(D_8\\), and an elementary abelian group of order 8. These results make symmetry-based searches finite in several branches. The trivial-group branch receives no comparable reduction.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "the full permutation automorphism group of every binary self-dual [72,36,16] code, if one exists"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1016/j.ffa.2013.07.007",
      "locator": "Martino Borello, Finite Fields and Their Applications 25 (2014), 1-7, Theorem 1.1 and final exclusion; combined with Vassil Yorgov and Daniel Yorgov, IEEE Transactions on Information Theory 60(6) (2014), 3302-3307, DOI 10.1109/TIT.2014.2313697"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1016/j.ffa.2013.07.007",
    "locator": "Martino Borello, Finite Fields and Their Applications 25 (2014), 1-7, Theorem 1.1 and final exclusion; combined with Vassil Yorgov and Daniel Yorgov, IEEE Transactions on Information Theory 60(6) (2014), 3302-3307, DOI 10.1109/TIT.2014.2313697"
  },
  "relations": [
    {
      "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"
    }
  ]
}