{"abstract":"Patterns with errors are declared decodable beyond the code capability.","category":"Error-correcting codes","checks":8,"contract":"Given a block code as a list of equal-length codewords (duplicates allowed but ignored; fewer than two distinct words or unequal lengths return None), compute the minimum Hamming distance d, the guaranteed correction capability t = floor((d-1)/2), detection capability d-1, and whether a pattern of the given errors and erasures is decodable (2*errors + erasures < d). Return [d, t, detect, decodable].","evaluation_group":"w2-error_correcting_codes-code-capability","failed_approach":"Using <= admits patterns that reach the distance exactly.","family":"w2-error_correcting_codes-code-capability-errors-erasures-bound","id":"FA-72121","implementations":{"attempt":{"sha256":"dba6f2ce34d11c2493cadf33543c34f3fb9b216341d822c2e29d286e43656ed0","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(codewords, errors, erasures):\n    words = [tuple(w) for w in codewords]\n    if len(set(words)) < 2 or len({len(w) for w in words}) != 1:\n        return None\n    uniq = sorted(set(words))\n    d = min(sum(a != b for a, b in zip(u, v)) for i, u in enumerate(uniq) for v in uniq[i + 1:])\n    return [d, (d - 1) // 2, d - 1, 2 * errors + erasures <= d]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['regression [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0], [1, 1, 1, 1, 1, 1]], 1, 0], [2, 0, 1, False]], ['regression [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0]], 1, 2], [4, 1, 3, False]], ['control [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1]], 0, 0]', [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1]], 0, 0], [1, 0, 0, True]], ['control [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1], [1, 0, 1, 0, 1...', [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1], [1, 0, 1, 0, 1]], 1, 2], [1, 0, 0, False]], ['control [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0]], 2, 3], [2, 0, 1, False]], ['control [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0]], 0, 2], [4, 1, 3, True]], ['control [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, ...', [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, 0, 1, 0, 0], [1, 0, 1, 0, 0, 1, 0, 1]], 0, 3], [2, 0, 1, False]], ['control [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, ...', [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, 0, 1, 0, 0], [1, 0, 1, 0, 0, 1, 0, 1], [0, 1, 1, 1, 1, 1, 1, 1]], 0, 0], [2, 0, 1, True]]], [['regression [[[0, 0, 0], [1, 1, 1]], 1, 1]', [[[0, 0, 0], [1, 1, 1]], 1, 1], [3, 1, 2, False]], ['regression [[[0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 1], [0, 1, 0, 1, 0, 1, ...', [[[0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 1], [0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0, 0], [0, 1, 0, 0, 1, 0, 1], [1, 1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1, 1], [1, 1, 1, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0, 1], [1, 0, 1, 1, 0, 1, 0], [0, 1, 1, 0, 0, 1, 1], [0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 1, 0, 1], [0, 0, 1, 0, 1, 1, 0], [1, 1, 1, 1, 1, 1, 1]], 1, 1], [3, 1, 2, False]], ['partial-repair [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1], [5, 2, 4, False]], ['control [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, ...', [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, 0, 1, 0, 0], [1, 0, 1, 0, 0, 1, 0, 1], [0, 1, 1, 1, 1, 1, 1, 1]], 0, 0], [2, 0, 1, True]], ['control [[[0, 1, 0, 0, 1], [0, 0, 0, 0, 1]], 1, 3]', [[[0, 1, 0, 0, 1], [0, 0, 0, 0, 1]], 1, 3], [1, 0, 0, False]], ['control [[[0, 1, 0, 0, 1], [0, 0, 0, 0, 1], [0, 1, 0, 0, 1]], 1, 0]', [[[0, 1, 0, 0, 1], [0, 0, 0, 0, 1], [0, 1, 0, 0, 1]], 1, 0], [1, 0, 0, False]], ['control [[[0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1, ...', [[[0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1, 1], [1, 1, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0]], 0, 3], [1, 0, 0, False]], ['control [[[0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1, ...', [[[0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1, 1], [1, 1, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0]], 1, 2], [1, 0, 0, False]]], [['regression [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0]], 1, 2], [4, 1, 3, False]], ['regression [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1], [5, 2, 4, False]], ['partial-repair [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0], [1, 1, 1, 1, 1, 1]], 1, 0], [2, 0, 1, False]], ['control [[[0, 1, 1, 0, 0, 1, 0, 1, 0], [1, 1, 1, 1, 1, 1, 0, 0, 1], [0, 1, ...', [[[0, 1, 1, 0, 0, 1, 0, 1, 0], [1, 1, 1, 1, 1, 1, 0, 0, 1], [0, 1, 1, 0, 1, 1, 1, 1, 0], [0, 1, 1, 0, 0, 1, 0, 1, 0]], 1, 2], [2, 0, 1, False]], ['control [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 1, 1]], 1, 0]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 1, 1]], 1, 0], [5, 2, 4, True]], ['control [[[0, 0, 0], [1, 1, 1]], 1, 0]', [[[0, 0, 0], [1, 1, 1]], 1, 0], [3, 1, 2, True]], ['control [[[0, 0, 0], [1, 1, 1]], 0, 2]', [[[0, 0, 0], [1, 1, 1]], 0, 2], [3, 1, 2, True]], ['control [[[0, 0, 0, 0], [1, 1, 1, 1]], 1, 1]', [[[0, 0, 0, 0], [1, 1, 1, 1]], 1, 1], [4, 1, 3, True]]], [['regression [[[0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 1], [0, 1, 0, 1, 0, 1, ...', [[[0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 1], [0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0, 0], [0, 1, 0, 0, 1, 0, 1], [1, 1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1, 1], [1, 1, 1, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0, 1], [1, 0, 1, 1, 0, 1, 0], [0, 1, 1, 0, 0, 1, 1], [0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 1, 0, 1], [0, 0, 1, 0, 1, 1, 0], [1, 1, 1, 1, 1, 1, 1]], 1, 1], [3, 1, 2, False]], ['regression [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0], [1, 1, 1, 1, 1, 1]], 1, 0], [2, 0, 1, False]], ['partial-repair [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1], [5, 2, 4, False]], ['partial-repair [[[0, 0, 0], [1, 1, 1]], 1, 1]', [[[0, 0, 0], [1, 1, 1]], 1, 1], [3, 1, 2, False]], ['control [[[0, 0, 0, 0], [1, 1, 1, 1]], 0, 3]', [[[0, 0, 0, 0], [1, 1, 1, 1]], 0, 3], [4, 1, 3, True]], ['control [[[0, 0, 0, 0, 0], [1, 1, 1, 1, 1], [0, 0, 0, 0, 0]], 2, 0]', [[[0, 0, 0, 0, 0], [1, 1, 1, 1, 1], [0, 0, 0, 0, 0]], 2, 0], [5, 2, 4, True]], ['control [[[0, 1], [0, 1]], 0, 0]', [[[0, 1], [0, 1]], 0, 0], None], ['control [[[0, 1], [1, 0, 1]], 0, 0]', [[[0, 1], [1, 0, 1]], 0, 0], None]], [['regression [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1], [5, 2, 4, False]], ['regression [[[0, 0, 0], [1, 1, 1]], 1, 1]', [[[0, 0, 0], [1, 1, 1]], 1, 1], [3, 1, 2, False]], ['partial-repair [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0], [1, 1, 1, 1, 1, 1]], 1, 0], [2, 0, 1, False]], ['partial-repair [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0]], 1, 2], [4, 1, 3, False]], ['control [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1]], 0, 0]', [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1]], 0, 0], [1, 0, 0, True]], ['control [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1], [1, 0, 1, 0, 1...', [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1], [1, 0, 1, 0, 1]], 1, 2], [1, 0, 0, False]], ['control [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0]], 2, 3], [2, 0, 1, False]], ['control [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0]], 0, 2], [4, 1, 3, True]]]]\nfor label, args, expected in fixtures[N - 1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"498ab587927bce026395cedb01217419658b618343e4c7cf486e83886d9a8496","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(codewords, errors, erasures):\n    words = [tuple(w) for w in codewords]\n    if len(set(words)) < 2 or len({len(w) for w in words}) != 1:\n        return None\n    uniq = sorted(set(words))\n    d = min(sum(a != b for a, b in zip(u, v)) for i, u in enumerate(uniq) for v in uniq[i + 1:])\n    return [d, (d - 1) // 2, d - 1, errors + erasures < d]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['regression [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0], [1, 1, 1, 1, 1, 1]], 1, 0], [2, 0, 1, False]], ['regression [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0]], 1, 2], [4, 1, 3, False]], ['control [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1]], 0, 0]', [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1]], 0, 0], [1, 0, 0, True]], ['control [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1], [1, 0, 1, 0, 1...', [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1], [1, 0, 1, 0, 1]], 1, 2], [1, 0, 0, False]], ['control [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0]], 2, 3], [2, 0, 1, False]], ['control [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0]], 0, 2], [4, 1, 3, True]], ['control [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, ...', [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, 0, 1, 0, 0], [1, 0, 1, 0, 0, 1, 0, 1]], 0, 3], [2, 0, 1, False]], ['control [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, ...', [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, 0, 1, 0, 0], [1, 0, 1, 0, 0, 1, 0, 1], [0, 1, 1, 1, 1, 1, 1, 1]], 0, 0], [2, 0, 1, True]]], [['regression [[[0, 0, 0], [1, 1, 1]], 1, 1]', [[[0, 0, 0], [1, 1, 1]], 1, 1], [3, 1, 2, False]], ['regression [[[0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 1], [0, 1, 0, 1, 0, 1, ...', [[[0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 1], [0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0, 0], [0, 1, 0, 0, 1, 0, 1], [1, 1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1, 1], [1, 1, 1, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0, 1], [1, 0, 1, 1, 0, 1, 0], [0, 1, 1, 0, 0, 1, 1], [0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 1, 0, 1], [0, 0, 1, 0, 1, 1, 0], [1, 1, 1, 1, 1, 1, 1]], 1, 1], [3, 1, 2, False]], ['partial-repair [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1], [5, 2, 4, False]], ['control [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, ...', [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, 0, 1, 0, 0], [1, 0, 1, 0, 0, 1, 0, 1], [0, 1, 1, 1, 1, 1, 1, 1]], 0, 0], [2, 0, 1, True]], ['control [[[0, 1, 0, 0, 1], [0, 0, 0, 0, 1]], 1, 3]', [[[0, 1, 0, 0, 1], [0, 0, 0, 0, 1]], 1, 3], [1, 0, 0, False]], ['control [[[0, 1, 0, 0, 1], [0, 0, 0, 0, 1], [0, 1, 0, 0, 1]], 1, 0]', [[[0, 1, 0, 0, 1], [0, 0, 0, 0, 1], [0, 1, 0, 0, 1]], 1, 0], [1, 0, 0, False]], ['control [[[0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1, ...', [[[0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1, 1], [1, 1, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0]], 0, 3], [1, 0, 0, False]], ['control [[[0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1, ...', [[[0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1, 1], [1, 1, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0]], 1, 2], [1, 0, 0, False]]], [['regression [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0]], 1, 2], [4, 1, 3, False]], ['regression [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1], [5, 2, 4, False]], ['partial-repair [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0], [1, 1, 1, 1, 1, 1]], 1, 0], [2, 0, 1, False]], ['control [[[0, 1, 1, 0, 0, 1, 0, 1, 0], [1, 1, 1, 1, 1, 1, 0, 0, 1], [0, 1, ...', [[[0, 1, 1, 0, 0, 1, 0, 1, 0], [1, 1, 1, 1, 1, 1, 0, 0, 1], [0, 1, 1, 0, 1, 1, 1, 1, 0], [0, 1, 1, 0, 0, 1, 0, 1, 0]], 1, 2], [2, 0, 1, False]], ['control [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 1, 1]], 1, 0]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 1, 1]], 1, 0], [5, 2, 4, True]], ['control [[[0, 0, 0], [1, 1, 1]], 1, 0]', [[[0, 0, 0], [1, 1, 1]], 1, 0], [3, 1, 2, True]], ['control [[[0, 0, 0], [1, 1, 1]], 0, 2]', [[[0, 0, 0], [1, 1, 1]], 0, 2], [3, 1, 2, True]], ['control [[[0, 0, 0, 0], [1, 1, 1, 1]], 1, 1]', [[[0, 0, 0, 0], [1, 1, 1, 1]], 1, 1], [4, 1, 3, True]]], [['regression [[[0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 1], [0, 1, 0, 1, 0, 1, ...', [[[0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 1], [0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0, 0], [0, 1, 0, 0, 1, 0, 1], [1, 1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1, 1], [1, 1, 1, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0, 1], [1, 0, 1, 1, 0, 1, 0], [0, 1, 1, 0, 0, 1, 1], [0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 1, 0, 1], [0, 0, 1, 0, 1, 1, 0], [1, 1, 1, 1, 1, 1, 1]], 1, 1], [3, 1, 2, False]], ['regression [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0], [1, 1, 1, 1, 1, 1]], 1, 0], [2, 0, 1, False]], ['partial-repair [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1], [5, 2, 4, False]], ['partial-repair [[[0, 0, 0], [1, 1, 1]], 1, 1]', [[[0, 0, 0], [1, 1, 1]], 1, 1], [3, 1, 2, False]], ['control [[[0, 0, 0, 0], [1, 1, 1, 1]], 0, 3]', [[[0, 0, 0, 0], [1, 1, 1, 1]], 0, 3], [4, 1, 3, True]], ['control [[[0, 0, 0, 0, 0], [1, 1, 1, 1, 1], [0, 0, 0, 0, 0]], 2, 0]', [[[0, 0, 0, 0, 0], [1, 1, 1, 1, 1], [0, 0, 0, 0, 0]], 2, 0], [5, 2, 4, True]], ['control [[[0, 1], [0, 1]], 0, 0]', [[[0, 1], [0, 1]], 0, 0], None], ['control [[[0, 1], [1, 0, 1]], 0, 0]', [[[0, 1], [1, 0, 1]], 0, 0], None]], [['regression [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1], [5, 2, 4, False]], ['regression [[[0, 0, 0], [1, 1, 1]], 1, 1]', [[[0, 0, 0], [1, 1, 1]], 1, 1], [3, 1, 2, False]], ['partial-repair [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0], [1, 1, 1, 1, 1, 1]], 1, 0], [2, 0, 1, False]], ['partial-repair [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0]], 1, 2], [4, 1, 3, False]], ['control [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1]], 0, 0]', [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1]], 0, 0], [1, 0, 0, True]], ['control [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1], [1, 0, 1, 0, 1...', [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1], [1, 0, 1, 0, 1]], 1, 2], [1, 0, 0, False]], ['control [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0]], 2, 3], [2, 0, 1, False]], ['control [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0]], 0, 2], [4, 1, 3, True]]]]\nfor label, args, expected in fixtures[N - 1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"fixed":{"sha256":"c961edec0bf25fdcb5e6ca4c5eb3781161a71a55192d4fc70a8f4a7b29ee9789","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(codewords, errors, erasures):\n    words = [tuple(w) for w in codewords]\n    if len(set(words)) < 2 or len({len(w) for w in words}) != 1:\n        return None\n    uniq = sorted(set(words))\n    d = min(sum(a != b for a, b in zip(u, v)) for i, u in enumerate(uniq) for v in uniq[i + 1:])\n    return [d, (d - 1) // 2, d - 1, 2 * errors + erasures < d]\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['regression [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0], [1, 1, 1, 1, 1, 1]], 1, 0], [2, 0, 1, False]], ['regression [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0]], 1, 2], [4, 1, 3, False]], ['control [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1]], 0, 0]', [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1]], 0, 0], [1, 0, 0, True]], ['control [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1], [1, 0, 1, 0, 1...', [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1], [1, 0, 1, 0, 1]], 1, 2], [1, 0, 0, False]], ['control [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0]], 2, 3], [2, 0, 1, False]], ['control [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0]], 0, 2], [4, 1, 3, True]], ['control [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, ...', [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, 0, 1, 0, 0], [1, 0, 1, 0, 0, 1, 0, 1]], 0, 3], [2, 0, 1, False]], ['control [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, ...', [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, 0, 1, 0, 0], [1, 0, 1, 0, 0, 1, 0, 1], [0, 1, 1, 1, 1, 1, 1, 1]], 0, 0], [2, 0, 1, True]]], [['regression [[[0, 0, 0], [1, 1, 1]], 1, 1]', [[[0, 0, 0], [1, 1, 1]], 1, 1], [3, 1, 2, False]], ['regression [[[0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 1], [0, 1, 0, 1, 0, 1, ...', [[[0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 1], [0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0, 0], [0, 1, 0, 0, 1, 0, 1], [1, 1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1, 1], [1, 1, 1, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0, 1], [1, 0, 1, 1, 0, 1, 0], [0, 1, 1, 0, 0, 1, 1], [0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 1, 0, 1], [0, 0, 1, 0, 1, 1, 0], [1, 1, 1, 1, 1, 1, 1]], 1, 1], [3, 1, 2, False]], ['partial-repair [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1], [5, 2, 4, False]], ['control [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, ...', [[[0, 1, 1, 1, 1, 1, 1, 1], [0, 1, 0, 0, 0, 1, 1, 1], [0, 0, 1, 0, 0, 1, 0, 0], [1, 0, 1, 0, 0, 1, 0, 1], [0, 1, 1, 1, 1, 1, 1, 1]], 0, 0], [2, 0, 1, True]], ['control [[[0, 1, 0, 0, 1], [0, 0, 0, 0, 1]], 1, 3]', [[[0, 1, 0, 0, 1], [0, 0, 0, 0, 1]], 1, 3], [1, 0, 0, False]], ['control [[[0, 1, 0, 0, 1], [0, 0, 0, 0, 1], [0, 1, 0, 0, 1]], 1, 0]', [[[0, 1, 0, 0, 1], [0, 0, 0, 0, 1], [0, 1, 0, 0, 1]], 1, 0], [1, 0, 0, False]], ['control [[[0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1, ...', [[[0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1, 1], [1, 1, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0]], 0, 3], [1, 0, 0, False]], ['control [[[0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1, ...', [[[0, 1, 0, 0, 0, 0, 0], [0, 1, 1, 0, 1, 1, 1], [0, 1, 1, 0, 0, 1, 1], [1, 1, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0, 0]], 1, 2], [1, 0, 0, False]]], [['regression [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0]], 1, 2], [4, 1, 3, False]], ['regression [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1], [5, 2, 4, False]], ['partial-repair [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0], [1, 1, 1, 1, 1, 1]], 1, 0], [2, 0, 1, False]], ['control [[[0, 1, 1, 0, 0, 1, 0, 1, 0], [1, 1, 1, 1, 1, 1, 0, 0, 1], [0, 1, ...', [[[0, 1, 1, 0, 0, 1, 0, 1, 0], [1, 1, 1, 1, 1, 1, 0, 0, 1], [0, 1, 1, 0, 1, 1, 1, 1, 0], [0, 1, 1, 0, 0, 1, 0, 1, 0]], 1, 2], [2, 0, 1, False]], ['control [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 1, 1]], 1, 0]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 1, 1]], 1, 0], [5, 2, 4, True]], ['control [[[0, 0, 0], [1, 1, 1]], 1, 0]', [[[0, 0, 0], [1, 1, 1]], 1, 0], [3, 1, 2, True]], ['control [[[0, 0, 0], [1, 1, 1]], 0, 2]', [[[0, 0, 0], [1, 1, 1]], 0, 2], [3, 1, 2, True]], ['control [[[0, 0, 0, 0], [1, 1, 1, 1]], 1, 1]', [[[0, 0, 0, 0], [1, 1, 1, 1]], 1, 1], [4, 1, 3, True]]], [['regression [[[0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 1], [0, 1, 0, 1, 0, 1, ...', [[[0, 0, 0, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 1], [0, 1, 0, 1, 0, 1, 0], [1, 0, 0, 0, 0, 1, 1], [1, 0, 0, 1, 1, 0, 0], [0, 1, 0, 0, 1, 0, 1], [1, 1, 0, 0, 1, 1, 0], [0, 0, 0, 1, 1, 1, 1], [1, 1, 1, 0, 0, 0, 0], [0, 0, 1, 1, 0, 0, 1], [1, 0, 1, 1, 0, 1, 0], [0, 1, 1, 0, 0, 1, 1], [0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 1, 0, 1], [0, 0, 1, 0, 1, 1, 0], [1, 1, 1, 1, 1, 1, 1]], 1, 1], [3, 1, 2, False]], ['regression [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0], [1, 1, 1, 1, 1, 1]], 1, 0], [2, 0, 1, False]], ['partial-repair [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1], [5, 2, 4, False]], ['partial-repair [[[0, 0, 0], [1, 1, 1]], 1, 1]', [[[0, 0, 0], [1, 1, 1]], 1, 1], [3, 1, 2, False]], ['control [[[0, 0, 0, 0], [1, 1, 1, 1]], 0, 3]', [[[0, 0, 0, 0], [1, 1, 1, 1]], 0, 3], [4, 1, 3, True]], ['control [[[0, 0, 0, 0, 0], [1, 1, 1, 1, 1], [0, 0, 0, 0, 0]], 2, 0]', [[[0, 0, 0, 0, 0], [1, 1, 1, 1, 1], [0, 0, 0, 0, 0]], 2, 0], [5, 2, 4, True]], ['control [[[0, 1], [0, 1]], 0, 0]', [[[0, 1], [0, 1]], 0, 0], None], ['control [[[0, 1], [1, 0, 1]], 0, 0]', [[[0, 1], [1, 0, 1]], 0, 0], None]], [['regression [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1]', [[[1, 1, 0, 1, 1, 1], [0, 0, 0, 0, 0, 0]], 2, 1], [5, 2, 4, False]], ['regression [[[0, 0, 0], [1, 1, 1]], 1, 1]', [[[0, 0, 0], [1, 1, 1]], 1, 1], [3, 1, 2, False]], ['partial-repair [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0], [1, 1, 1, 1, 1, 1]], 1, 0], [2, 0, 1, False]], ['partial-repair [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0]], 1, 2], [4, 1, 3, False]], ['control [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1]], 0, 0]', [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1]], 0, 0], [1, 0, 0, True]], ['control [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1], [1, 0, 1, 0, 1...', [[[1, 0, 1, 0, 1], [1, 1, 0, 1, 1], [1, 1, 1, 1, 1], [1, 0, 1, 0, 1]], 1, 2], [1, 0, 0, False]], ['control [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0...', [[[1, 1, 1, 1, 1, 1], [1, 1, 0, 0, 0, 0], [1, 0, 1, 0, 0, 1], [1, 0, 1, 1, 1, 0]], 2, 3], [2, 0, 1, False]], ['control [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, ...', [[[0, 1, 1, 1, 1, 0, 0], [1, 0, 1, 0, 0, 1, 1], [1, 1, 0, 0, 0, 0, 0]], 0, 2], [4, 1, 3, True]]]]\nfor label, args, expected in fixtures[N - 1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":"A deterministic, bounded teaching model of the named code under the stated contract; not a production codec. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"w2-error_correcting_codes-code-capability-errors-erasures-bound","generated_at":"2026-09-29T14:48:35.675540+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Link and storage designers budget error and erasure correction from the minimum distance.","repair":"Require 2 * errors + erasures < d.","root_cause":"The bound is errors + erasures < d; each unknown-position error costs two units of distance.","sha256":"a1627afa53ecb3f018141693099603730b325e22646f1554ee7c1134d74ebfd4","title":"Code capability weighs errors like erasures · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary 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