{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R584",
  "content_hash": null,
  "slug": "pirw600-claim-certified-weight-interval",
  "type": "claim",
  "title": "The minimum tap weight lies between 8 and 186",
  "summary": "Exact replay proves feasibility at weight 186 and excludes every candidate of weight at most 7.",
  "relevance": "For Sparsest degree-600 recurrence for a prime-indicator prefix, record pirw600-claim-certified-weight-interval (“The minimum tap weight lies between 8 and 186”) records a bound, answer, status fact, or structural consequence. The record states: Exact replay proves feasibility at weight 186 and excludes every candidate of weight at most 7.",
  "relevance_source": "recorded",
  "body": "Write\n\\[\n v_j=(s_j,s_{j+1},\\ldots,s_{j+423})\\in\\mathbf F_2^{424}.\n\\]\nSince the endpoint coefficients are fixed at one, an internal tap set \\(T\\subseteq\\{1,\\ldots,599\\}\\) is feasible exactly when\n\\[\n \\bigoplus_{j\\in T}v_j=v_0\\oplus v_{600}.\n\\]\nThe exact verifier forms all 599 internal columns and all 179,101 pairwise xor values. It tests internal subsets of sizes zero through four with singleton and pair tables. For size five, it checks all 35,641,099 triples against the pair table. If the two sides of one of these table identities share an index, cancellation gives a smaller solution already excluded. The sweep finds no solution with at most five internal taps. Every admissible vector has two endpoint taps, so its total weight is at least 8.\n\nFor the upper bound, interpret the hexadecimal integer\n```\n104020000042401000000000000100000010004020001702093a81120945a122179a133c52f6f5d7860361026a5b18b00f8b704122171b46d22ccedce3191b108a19415c43e921a850ae209\n```\nwith coefficient \\(c_j\\) in bit \\(j\\), starting at the least significant bit. It has 601 bits, both endpoint bits are one, and its Hamming weight is 186. Exact substitution gives zero violations among all 424 recurrence equations.\n\nRow reduction of the 424 recurrence rows together with the two endpoint rows gives rank 426 and affine dimension 175. The exact minimum inside the certified interval\n\\[\n8\\leq w_{\\min}\\leq186\n\\]\nremains open. A two-minute pseudo-Boolean solver probe at threshold 150 ended without a model or an unsatisfiability proof and contributes no further bound.",
  "status": "supported",
  "evidence_grade": "computational",
  "scope": {
    "kind": "bounded",
    "statement": "the stated 601 binary coefficients, 424 recurrence equations, and prime-indicator prefix of length 1024",
    "bounds": {
      "degree": {
        "min": 600,
        "max": 600
      },
      "prefix_length": {
        "min": 1024,
        "max": 1024
      },
      "recurrence_equations": {
        "min": 424,
        "max": 424
      }
    },
    "exhaustive": false
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1109/TIT.1969.1054260",
      "locator": "Exact standard-library computation in pirw600-artifact-subset-and-incumbent-certificate, executed on 2026-07-25"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1109/TIT.1969.1054260",
    "locator": "Exact standard-library computation in pirw600-artifact-subset-and-incumbent-certificate, executed on 2026-07-25"
  },
  "relations": [
    {
      "slug": "R582",
      "title": "Exact lower-bound and incumbent replay certificate",
      "object_type": "artifact",
      "relation": "supports",
      "direction": "incoming"
    },
    {
      "slug": "R583",
      "title": "The fixed-degree tap objective is a syndrome-decoding instance",
      "object_type": "attempt",
      "relation": "contextualizes",
      "direction": "incoming"
    },
    {
      "slug": "prime-indicator-recurrence-weight-600",
      "title": "prime indicator recurrence weight 600",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}