{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R623",
  "content_hash": null,
  "slug": "qafp-code-bound",
  "type": "claim",
  "title": "The current coding interval is 333 through 381, with a gap below the design endpoint",
  "summary": "An explicit 333-plane code supplies the lower bound, while divisibility and extension results imply that the maximum is at most 378 unless it equals 381.",
  "relevance": "For A binary q-analog of the Fano plane, record qafp-code-bound (“The current coding interval is 333 through 381, with a gap below the design endpoint”) records a bound, answer, status fact, or structural consequence. The record states: An explicit 333-plane code supplies the lower bound, while divisibility and extension results imply that the maximum is at most 378 unless it equals 381.",
  "relevance_source": "recorded",
  "body": "Heinlein, Kiermaier, Kurz, and Wassermann constructed an explicit 333-plane code, proving\n\\[\nA_2(7,4;3)\\geq333.\n\\]\nThe line-packing bound gives \\(A_2(7,4;3)\\leq381\\). The 2025 survey records a sharper alternative obtained from extendability and divisible-code results:\n\\[\nA_2(7,4;3)\\leq378\\quad\\text{or}\\quad A_2(7,4;3)=381.\n\\]\nA code at the second endpoint is exactly a binary q-Fano plane. The known 333-plane construction therefore measures progress on a relaxation. An extension certificate would still be needed to turn it into a solution of the 381-block exact cover.",
  "status": "supported",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "the maximum size A_2(7,4;3) of a binary 3-dimensional constant-dimension code in ambient dimension 7 with minimum subspace distance 4",
    "bounds": {
      "field_order": {
        "min": 2,
        "max": 2
      },
      "ambient_dimension": {
        "min": 7,
        "max": 7
      },
      "subspace_dimension": {
        "min": 3,
        "max": 3
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.3934/amc.2019029",
      "locator": "Heinlein, Kiermaier, Kurz, and Wassermann, A subspace code of size 333 in the setting of a binary q-analog of the Fano plane, Theorem 2; Kurz 2025 survey, section 6, pages 76-79"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.3934/amc.2019029",
    "locator": "Heinlein, Kiermaier, Kurz, and Wassermann, A subspace code of size 333 in the setting of a binary q-analog of the Fano plane, Theorem 2; Kurz 2025 survey, section 6, pages 76-79"
  },
  "models": [],
  "relations": [
    {
      "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"
    }
  ]
}