{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R826",
  "content_hash": null,
  "slug": "vt0-claim-published-coefficient-formula",
  "type": "claim",
  "title": "A divisor sum gives every zero-syndrome weight coefficient",
  "summary": "The published VT weight-distribution formula reduces the bounded question to integer arithmetic over divisors of n+1.",
  "relevance": "This records a result, observation, or conjecture used to assess the target problem.",
  "relevance_source": "derived",
  "body": "Bibak, Kapron, and Srinivasan give the weight distribution of \\(VT_b(n)\\) in Corollary 3.12. Bibak and Milenkovic derive the same formula as Theorem IV.8 from their weight enumerator in Theorem IV.7. Setting \\(b=0\\) turns the Ramanujan sum \\(c_d(0)\\) into Euler's totient \\(\\varphi(d)\\), giving\n\\[\nW_{n,k}=\\frac{(-1)^k}{n+1}\\sum_{d\\mid n+1}(-1)^{\\lfloor k/d\\rfloor}\\varphi(d)\\binom{(n+1)/d-1}{\\lfloor k/d\\rfloor}.\n\\]\nEvery term is an integer expression. The artifact evaluates the numerator, checks divisibility by \\(n+1\\), and then tests the coefficient inequalities.",
  "status": "established",
  "evidence_grade": "sourced",
  "scope": {
    "kind": "universal",
    "statement": "every positive integer length n and every weight 0 <= k <= n for the zero-syndrome binary Varshamov-Tenengolts code"
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "source_only",
    "kind": "claim",
    "citation": {
      "url": "https://doi.org/10.1109/TCOMM.2018.2886354",
      "locator": "Khodakhast Bibak and Olgica Milenkovic, Explicit Formulas for the Weight Enumerators of Some Classes of Deletion Correcting Codes, IEEE Transactions on Communications 67(3) (2019), 1809-1816, Theorems IV.7 and IV.8; Khodakhast Bibak, Bruce M. Kapron, and Venkatesh Srinivasan, Unweighted Linear Congruences with Distinct Coordinates and the Varshamov-Tenengolts Codes, Designs, Codes and Cryptography 86 (2018), 1893-1904, Corollary 3.12, DOI 10.1007/s10623-017-0428-3"
    },
    "missing": [
      "source",
      "command",
      "runtime",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://doi.org/10.1109/TCOMM.2018.2886354",
    "locator": "Khodakhast Bibak and Olgica Milenkovic, Explicit Formulas for the Weight Enumerators of Some Classes of Deletion Correcting Codes, IEEE Transactions on Communications 67(3) (2019), 1809-1816, Theorems IV.7 and IV.8; Khodakhast Bibak, Bruce M. Kapron, and Venkatesh Srinivasan, Unweighted Linear Congruences with Distinct Coordinates and the Varshamov-Tenengolts Codes, Designs, Codes and Cryptography 86 (2018), 1893-1904, Corollary 3.12, DOI 10.1007/s10623-017-0428-3"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R827",
      "title": "The truncated VT zero-syndrome weight sequence is log-concave through length 2000",
      "object_type": "claim",
      "relation": "supports",
      "direction": "outgoing"
    },
    {
      "slug": "R824",
      "title": "Replayable exact coefficient sweep through length 2000",
      "object_type": "artifact",
      "relation": "uses",
      "direction": "incoming"
    },
    {
      "slug": "vt-zero-weight-log-concavity",
      "title": "vt zero weight log concavity",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}