{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R753",
  "content_hash": null,
  "slug": "ssm34-claim-goethals-seidel-lower-bound",
  "type": "claim",
  "title": "An explicit order-34 matrix has determinant 35^16",
  "summary": "Deleting two matching rows and columns from the Goethals-Seidel skew conference matrix of order 36 gives Pfaffian -35^8.",
  "relevance": "For Maximum determinant of a skew Seidel matrix of order 34, record ssm34-claim-goethals-seidel-lower-bound (“An explicit order-34 matrix has determinant 35^16”) records a bound, answer, status fact, or structural consequence. The record states: Deleting two matching rows and columns from the Goethals-Seidel skew conference matrix of order 36 gives Pfaffian -35^8.",
  "relevance_source": "recorded",
  "body": "Goethals and Seidel give four circulant first rows of length nine:\n\\[\n\\begin{aligned}\na&=(0,+,+,-,+,-,+,-,-),\\\\\nb&=(+,-,+,+,-,-,+,+,-),\\\\\nc&=(-,-,+,+,+,+,+,+,-),\\\\\nd&=(+,+,+,-,+,+,-,+,+).\n\\end{aligned}\n\\]\nLet \\(A,B,C,D\\) be the corresponding circulants, let \\(R\\) be the reversal matrix, and use their four-by-four block array. The replay verifies that the resulting \\(H\\) has entries in \\(\\{-1,1\\}\\), satisfies \\(H+H^T=2I\\), and obeys \\(HH^T=36I\\). Thus \\(K=H-I\\) is a skew conference matrix with \\(KK^T=35I\\).\n\nDelete rows and columns 0 and 1 from \\(K\\). The remaining matrix \\(S\\) is a skew Seidel matrix of order 34. Exact Pfaffian elimination gives\n\\[\n\\operatorname{pf}(S)=-2{,}251{,}875{,}390{,}625=-35^8,\n\\]\nand fraction-free Bareiss elimination gives\n\\[\n\\det S=5{,}070{,}942{,}774{,}902{,}496{,}337{,}890{,}625=35^{16}.\n\\]\nThis is also the order-34 case of the principal-submatrix lower bound in Theorem 5.5 of Klanderman et al.",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "the order-34 principal submatrix obtained by deleting indices 0 and 1 from the Goethals-Seidel order-36 skew conference matrix",
    "bounds": {
      "source_order": {
        "min": 36,
        "max": 36
      },
      "principal_submatrix_order": {
        "min": 34,
        "max": 34
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1017/S144678870000673X",
      "locator": "Goethals and Seidel, A skew Hadamard matrix of order 36, Theorems 1 and 2, pages 343-344; exact replay in ssm34-artifact-gs36-principal-minor"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1017/S144678870000673X",
    "locator": "Goethals and Seidel, A skew Hadamard matrix of order 36, Theorems 1 and 2, pages 343-344; exact replay in ssm34-artifact-gs36-principal-minor"
  },
  "relations": [
    {
      "slug": "R751",
      "title": "Exact generator, Pfaffian, and determinant verifier",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R752",
      "title": "The maximum is between 35^16 and 6,876,227,375,063^2",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "skew-seidel-maxdet-34",
      "title": "skew seidel maxdet 34",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}