{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R625",
  "content_hash": null,
  "slug": "qafp-open-status",
  "type": "claim",
  "title": "Existence of the binary q-Fano plane remains open",
  "summary": "The latest primary survey located in this audit, dated 2025, still lists the 381-block case as unresolved.",
  "relevance": "For A binary q-analog of the Fano plane, record qafp-open-status (“Existence of the binary q-Fano plane remains open”) records a bound, answer, status fact, or structural consequence. The record states: The latest primary survey located in this audit, dated 2025, still lists the 381-block case as unresolved.",
  "relevance_source": "recorded",
  "body": "The question asks for a binary q-Steiner system \\(S_2(2,3,7)\\). Sascha Kurz's 2025 survey calls \\(A_2(7,4;3)\\) the smallest unknown binary constant-dimension-code value and says the q-Fano existence question remains widely open. The peer-reviewed 2022 paper of Michael Kiermaier likewise states that existence is undecided for every finite field order.\n\nFor the binary instance, a positive certificate is a list of 381 planes whose contained lines partition all 2,667 lines of \\(\\mathbb F_2^7\\). A negative certificate must cover the full exact-cover problem. Searches that prescribe a nontrivial automorphism leave the rigid case untouched, so their failures do not decide this claim.",
  "status": "open",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "bounded",
    "statement": "existence of a binary 2-(7,3,1)_2 subspace design in the 7-dimensional vector space over F_2",
    "bounds": {
      "field_order": {
        "min": 2,
        "max": 2
      },
      "ambient_dimension": {
        "min": 7,
        "max": 7
      },
      "block_dimension": {
        "min": 3,
        "max": 3
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.15495/EPub_UBT_00008787",
      "locator": "Sascha Kurz, Constructions and bounds for subspace codes, 2025, section 6, pages 76-79 of the manuscript; Michael Kiermaier, On alpha-points of q-analogs of the Fano plane, Designs, Codes and Cryptography 90 (2022), pages 1335-1345"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.15495/EPub_UBT_00008787",
    "locator": "Sascha Kurz, Constructions and bounds for subspace codes, 2025, section 6, pages 76-79 of the manuscript; Michael Kiermaier, On alpha-points of q-analogs of the Fano plane, Designs, Codes and Cryptography 90 (2022), pages 1335-1345"
  },
  "models": [],
  "relations": [
    {
      "slug": "R624",
      "title": "Every solution has 381 blocks and fixed local incidence counts",
      "object_type": "claim",
      "relation": "refines",
      "direction": "incoming"
    },
    {
      "slug": "R626",
      "title": "Any solution has at most one nonidentity automorphism and tightly fixed intersections",
      "object_type": "claim",
      "relation": "constrains",
      "direction": "incoming"
    },
    {
      "slug": "R623",
      "title": "The current coding interval is 333 through 381, with a gap below the design endpoint",
      "object_type": "claim",
      "relation": "refines",
      "direction": "incoming"
    },
    {
      "slug": "q-analog-fano-plane",
      "title": "q analog fano plane",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}