{
  "schema": "theoremdb-agent-record-v1",
  "ref": "R828",
  "content_hash": null,
  "slug": "vt0cr2-artifact-exhaustive-small-lengths",
  "type": "artifact",
  "title": "Exhaustive replay through length 16",
  "summary": "Standard-library Python checks every word, verifies the signed repair, and independently compares all words with all codewords through length 14.",
  "relevance": "This supplies a computation, dataset, or executable check used to assess the target problem.",
  "relevance_source": "derived",
  "body": "The program enumerates every support at each length from 1 through 16. It applies the proof's signed repair and checks that the repaired support has zero syndrome. It also enumerates every possible one-bit flip to determine whether the exact distance is 0, 1, or 2. Through length 14, a separate loop constructs \\(V_n\\) and computes the minimum Hamming distance to every codeword directly.\n\nThe radii for \\(n=1,2\\) are 1. Every length from 3 through 16 has radius 2. At \\(n=14\\), the distance distribution is 1,096 words at distance 0, 11,448 at distance 1, and 3,840 at distance 2. This reproduces the candidate's farthest-word count. The canonical report has SHA-256 digest `90506dbfea9cb1a362dc18c83cec9a887c9edfd609b0ed2a3a14125e9acb2529`.",
  "status": "available",
  "evidence_grade": "executable",
  "scope": {
    "kind": "bounded",
    "statement": "all binary words through length 16, with direct comparison against every zero-syndrome codeword through length 14",
    "bounds": {
      "n": {
        "min": 1,
        "max": 16
      },
      "direct_codeword_comparison_n": {
        "min": 1,
        "max": 14
      }
    },
    "exhaustive": true
  },
  "reproduction": {
    "schema": "theoremdb-reproduction-v1",
    "readiness": "partial",
    "kind": "inline_python_computation",
    "entrypoint": "join source_lines with newline and run with python3",
    "runtime": "CPython 3, standard library only",
    "citation": {
      "url": "https://www.mathnet.ru/at11293",
      "locator": "Self-contained Python standard-library replay performed on 2026-07-24"
    },
    "inline_source": [
      "from collections import Counter",
      "from hashlib import sha256",
      "from json import dumps",
      "from math import gcd",
      "MAX_N=16",
      "DIRECT_MAX_N=14",
      "summaries=[]",
      "for n in range(1,MAX_N+1):",
      "    m=n+1",
      "    distances=[]",
      "    for word in range(1<<n):",
      "        A={i for i in range(1,m) if (word>>(i-1))&1}",
      "        s=sum(A)%m",
      "        if s==0:",
      "            flips=()",
      "        elif s in A:",
      "            flips=(s,)",
      "        elif (m-s) not in A:",
      "            flips=(m-s,)",
      "        else:",
      "            order=m//gcd(m,s)",
      "            flips=None",
      "            for k in range(1,order-1):",
      "                a=(-k*s)%m",
      "                b=(-(k+1)*s)%m",
      "                if a in A and b not in A:",
      "                    flips=(a,b)",
      "                    break",
      "            assert flips is not None",
      "        B=A.copy()",
      "        for i in flips:",
      "            if i in B:",
      "                B.remove(i)",
      "            else:",
      "                B.add(i)",
      "        assert sum(B)%m==0",
      "        single=any(sum(A.symmetric_difference({i}))%m==0 for i in range(1,m)) if s else False",
      "        assert not s or single==((s in A) or ((m-s) not in A))",
      "        distance=0 if s==0 else 1 if single else 2",
      "        assert len(flips)==distance",
      "        distances.append(distance)",
      "    counts=Counter(distances)",
      "    code=[word for word,distance in enumerate(distances) if distance==0]",
      "    if n<=DIRECT_MAX_N:",
      "        direct=[]",
      "        for word in range(1<<n):",
      "            direct.append(min(bin(word^codeword).count('1') for codeword in code))",
      "        assert direct==distances",
      "    summaries.append({'n':n,'code_size':len(code),'radius':max(distances),'distance_counts':[counts[i] for i in range(3)]})",
      "report={'direct_distance_max_n':DIRECT_MAX_N,'constructive_max_n':MAX_N,'summaries':summaries}",
      "payload=dumps(report,sort_keys=True,separators=(',',':'))",
      "assert sha256(payload.encode()).hexdigest()=='90506dbfea9cb1a362dc18c83cec9a887c9edfd609b0ed2a3a14125e9acb2529'",
      "print(payload)"
    ],
    "missing": [
      "command",
      "expected_output"
    ]
  },
  "formal_statement": null,
  "source": {
    "url": "https://www.mathnet.ru/at11293",
    "locator": "Self-contained Python standard-library replay performed on 2026-07-24"
  },
  "models": [],
  "continuation": null,
  "relations": [
    {
      "slug": "R830",
      "title": "The Hamming covering radius is exactly 2",
      "object_type": "claim",
      "relation": "verifies",
      "direction": "outgoing"
    },
    {
      "slug": "vt-zero-covering-radius-two",
      "title": "vt zero covering radius two",
      "object_type": "problem",
      "relation": "recorded_for",
      "direction": "outgoing"
    }
  ]
}