{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R503",
  "content_hash": null,
  "slug": "mdsc-claim-published-code-maximal-extension-334",
  "type": "claim",
  "title": "The published 333-plane code has a unique one-word extension",
  "summary": "Exact reconstruction and a scan of all 29,212 subspaces show that the whole space is the only new codeword compatible with every plane in the published code. Every supercode containing all 333 planes therefore has size at most 334.",
  "relevance": "For Exact mixed-dimension subspace-code number A_2(7,4), record mdsc-claim-published-code-maximal-extension-334 (“The published 333-plane code has a unique one-word extension”) records a bound, answer, status fact, or structural consequence. The record states: Exact reconstruction and a scan of all 29,212 subspaces show that the whole space is the only new codeword compatible with every plane in the published code.",
  "relevance_source": "recorded",
  "body": "The replay decodes the 103 representatives in Appendix C and generates their orbits under the printed Klein four-group. The orbit lengths are nine fixed planes, 26 pairs, and 68 four-element orbits, giving 333 distinct three-spaces. Their minimum distance is 4. Among every subspace of F₂⁷ outside this code, exactly one has distance at least 4 from all 333 planes: F₂⁷ itself. This is a maximal-extension result for this fixed construction. It supplies no global upper bound on A₂(7,4).",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "all supercodes obtained by retaining every word of the specific 333-plane Appendix C code and adjoining arbitrary subspaces of F_2^7",
    "bounds": {
      "starting_code_size": {
        "min": 333,
        "max": 333
      },
      "ambient_subspaces_checked": {
        "min": 29212,
        "max": 29212
      },
      "retained_starting_codewords": {
        "min": 333,
        "max": 333
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://arxiv.org/abs/1708.06224",
      "locator": "Heinlein et al., arXiv:1708.06224v5, group G_4,6 on PDF p. 14 and Appendix C on pp. 16-18; exact replay in mdsc-artifact-exact-f2-seven-census"
    },
    "runtime_seconds": 6.27,
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://arxiv.org/abs/1708.06224",
    "locator": "Heinlein et al., arXiv:1708.06224v5, group G_4,6 on PDF p. 14 and Appendix C on pp. 16-18; exact replay in mdsc-artifact-exact-f2-seven-census"
  },
  "relations": [
    {
      "slug": "R497",
      "title": "Exact F₂⁷ census, Appendix C replay, and Singer-orbit search",
      "object_type": "artifact",
      "relation": "evidences",
      "direction": "incoming"
    },
    {
      "slug": "R502",
      "title": "The dated interval is 334 ≤ A₂(7,4) ≤ 388",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "mixed-dimension-subspace-code-f2-7-d4",
      "title": "mixed dimension subspace code f2 7 d4",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}