files / problems / covering / share / 2026-08-24 / compute / witness_r10_n50.json

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{
  "name": "radius-two-r10-n50-q1",
  "field": "GF(2)",
  "redundancy": 10,
  "length": 50,
  "dimension": 40,
  "covering_radius": 2,
  "column_encoding": "Unsigned integers; bit i (least-significant bit first) is row i+1 of H.",
  "columns_decimal": [
    1,
    2,
    4,
    15,
    16,
    32,
    65,
    86,
    128,
    173,
    183,
    202,
    212,
    247,
    256,
    297,
    320,
    329,
    341,
    366,
    373,
    381,
    391,
    403,
    438,
    460,
    479,
    491,
    502,
    559,
    576,
    608,
    653,
    734,
    742,
    754,
    771,
    777,
    789,
    821,
    846,
    855,
    869,
    881,
    893,
    897,
    927,
    981,
    1003,
    1004
  ],
  "columns_hex": [
    "001", "002", "004", "00F", "010", "020", "041", "056", "080", "0AD",
    "0B7", "0CA", "0D4", "0F7", "100", "129", "140", "149", "155", "16E",
    "175", "17D", "187", "193", "1B6", "1CC", "1DF", "1EB", "1F6", "22F",
    "240", "260", "28D", "2DE", "2E6", "2F2", "303", "309", "315", "335",
    "34E", "357", "365", "371", "37D", "381", "39F", "3D5", "3EB", "3EC"
  ],
  "density": {
    "formula": "(1 + n + binom(n,2)) / 2^r",
    "numerator": 1276,
    "denominator": 1024,
    "reduced_numerator": 319,
    "reduced_denominator": 256,
    "decimal": 1.24609375
  },
  "documented_benchmark": {
    "redundancy": 10,
    "length": 51,
    "density_numerator": 1327,
    "density_denominator": 1024,
    "density_decimal": 1.2958984375,
    "source": "Davydov--Marcugini--Pambianco, New upper bounds for binary linear covering codes, Table 5.1 (arXiv:2511.02542v1, 2025-11-04)",
    "url": "https://arxiv.org/html/2511.02542"
  },
  "search": {
    "method": "fixed-cardinality targeted simulated annealing from the 51-column Kaikkonen--Rosendahl seed",
    "rng": "xorshift64, deterministic run 0",
    "accepted_witness_iteration": 3600281
  }
}