{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R624",
  "content_hash": null,
  "slug": "qafp-forced-parameters",
  "type": "claim",
  "title": "Every solution has 381 blocks and fixed local incidence counts",
  "summary": "There are 2,667 lines and each plane contains seven, forcing 381 blocks; every point, 5-space, and hyperplane then contains 21, 5, and 45 blocks.",
  "relevance": "For A binary q-analog of the Fano plane, record qafp-forced-parameters (“Every solution has 381 blocks and fixed local incidence counts”) records a bound, answer, status fact, or structural consequence. The record states: There are 2,667 lines and each plane contains seven, forcing 381 blocks; every point, 5-space, and hyperplane then contains 21, 5, and 45 blocks.",
  "relevance_source": "recorded",
  "body": "Gaussian-binomial evaluation gives\n\\[\n{7\\brack 1}_2=127,\\qquad {7\\brack 2}_2=2667,\\qquad {7\\brack 3}_2=11811,\n\\]\nand every 3-subspace contains \\({3\\brack2}_2=7\\) lines. Unique line coverage therefore forces\n\\[\n|\\mathcal B|=\\frac{{7\\brack2}_2}{{3\\brack2}_2}=381.\n\\]\nDouble counting gives 21 blocks through each point, 5 blocks in each 5-subspace, and 45 blocks in each hyperplane.\n\nThe equivalent exact-cover matrix has one column for each of the 2,667 lines and one row for each of the 11,811 planes. Every row has seven ones and every column has 31 ones. A set of 381 disjoint rows covering all columns would also be a constant-dimension code of size 381 and minimum subspace distance four. Thus the desired design exists exactly when \\(A_2(7,4;3)=381\\).",
  "status": "established",
  "evidence_grade": "reproduced",
  "scope": {
    "kind": "bounded",
    "statement": "Gaussian-binomial and incidence parameters forced by a binary 2-(7,3,1)_2 design",
    "bounds": {
      "field_order": {
        "min": 2,
        "max": 2
      },
      "ambient_dimension": {
        "min": 7,
        "max": 7
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.15495/EPub_UBT_00008787",
      "locator": "Kurz, Constructions and bounds for subspace codes, 2025, section 6; exact enumeration in qafp-artifact-incidence-replay"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.15495/EPub_UBT_00008787",
    "locator": "Kurz, Constructions and bounds for subspace codes, 2025, section 6; exact enumeration in qafp-artifact-incidence-replay"
  },
  "models": [],
  "relations": [
    {
      "slug": "R621",
      "title": "Exact enumeration of the 11,811 by 2,667 cover matrix",
      "object_type": "artifact",
      "relation": "verifies",
      "direction": "incoming"
    },
    {
      "slug": "R625",
      "title": "Existence of the binary q-Fano plane remains open",
      "object_type": "claim",
      "relation": "refines",
      "direction": "outgoing"
    },
    {
      "slug": "q-analog-fano-plane",
      "title": "q analog fano plane",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}