{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R662",
  "content_hash": null,
  "slug": "rm28-claim-certified-interval",
  "type": "claim",
  "title": "The full covering radius satisfies 88 <= rho(2,8) <= 96",
  "summary": "For length-256 truth tables, the best current certified interval is 88 <= rho(2,8) <= 96. The exact maximum distance over all 8-variable Boolean functions remains undetermined.",
  "relevance": "For Covering radius of the second-order Reed-Muller code RM(2,8), record rm28-claim-certified-interval (“The full covering radius satisfies 88 <= rho(2,8) <= 96”) records a bound, answer, status fact, or structural consequence. The record states: For length-256 truth tables, the best current certified interval is 88 <= rho(2,8) <= 96.",
  "relevance_source": "recorded",
  "body": "Write\n\\[\n\\rho(2,8)=\\max_{F:\\mathbb F_2^8\\to\\mathbb F_2}\\min_{Q\\in RM(2,8)}\\operatorname{wt}(F+Q).\n\\]\nThe cubic in rm28-claim-cubic-witness-distance-88 has distance 88 from \\(RM(2,8)\\), so \\(\\rho(2,8)\\geq 88\\). Wang proves \\(\\rho(2,7)=40\\) in Theorem 11 of the cited paper. The recursive inequality\n\\[\n\\rho(k,m)\\leq \\rho(k,m-1)+\\rho(k-1,m-1)\n\\]\nand the known value \\(\\rho(1,7)=56\\) give\n\\[\n\\rho(2,8)\\leq 40+56=96.\n\\]\nGillot and Langevin's current specialist page records the same interval and labels the second-order case in eight variables as open. A complete answer still needs either a global upper bound of 88 or an exhaustive classification that identifies a larger value in the interval.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the global covering radius of the second-order binary Reed-Muller code on eight variables",
    "bounds": {
      "order": {
        "min": 2,
        "max": 2
      },
      "variables": {
        "min": 8,
        "max": 8
      },
      "code_length": {
        "min": 256,
        "max": 256
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://langevin.univ-tln.fr/project/covering/covering.html",
      "locator": "Table of covering radii, row r(k,8), k=2; methodology and open-case statement, last modified February 2024"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://langevin.univ-tln.fr/project/covering/covering.html",
    "locator": "Table of covering radii, row r(k,8), k=2; methodology and open-case statement, last modified February 2024"
  },
  "relations": [
    {
      "slug": "R663",
      "title": "An eight-term cubic has exact second-order nonlinearity 88",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R664",
      "title": "The relative cubic covering radius equals 88",
      "object_type": "claim",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R660",
      "title": "A 2026-07-28 source audit confirms the current 88 to 96 interval",
      "object_type": "attempt",
      "relation": "informs",
      "direction": "incoming"
    },
    {
      "slug": "R659",
      "title": "Reproduce the B(3,4,7) high-nonlinearity orbit classification",
      "object_type": "attempt",
      "relation": "attempts",
      "direction": "incoming"
    },
    {
      "slug": "reed-muller-rm2-8-covering-radius",
      "title": "reed muller rm2 8 covering radius",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}