{
 "schema_version": "0.3",
 "name": "[[54,24,4]]",
 "code_type": "CSS",
 "n": 54,
 "k": 24,
 "checks": {
  "X": [
   [
    0,
    17,
    20,
    23,
    25,
    27,
    30,
    34,
    42,
    50
   ],
   [
    1,
    15,
    18,
    21,
    26,
    28,
    31,
    35,
    43,
    48
   ],
   [
    2,
    16,
    19,
    22,
    24,
    29,
    32,
    33,
    44,
    49
   ],
   [
    3,
    11,
    19,
    23,
    26,
    28,
    30,
    33,
    36,
    53
   ],
   [
    4,
    9,
    20,
    21,
    24,
    29,
    31,
    34,
    37,
    51
   ],
   [
    5,
    10,
    18,
    22,
    25,
    27,
    32,
    35,
    38,
    52
   ],
   [
    6,
    14,
    20,
    22,
    26,
    27,
    31,
    33,
    39,
    47
   ],
   [
    7,
    12,
    18,
    23,
    24,
    28,
    32,
    34,
    40,
    45
   ],
   [
    8,
    13,
    19,
    21,
    25,
    29,
    30,
    35,
    41,
    46
   ],
   [
    2,
    5,
    7,
    9,
    26,
    32,
    36,
    39,
    43,
    51
   ],
   [
    0,
    3,
    8,
    10,
    24,
    30,
    37,
    40,
    44,
    52
   ],
   [
    1,
    4,
    6,
    11,
    25,
    31,
    38,
    41,
    42,
    53
   ],
   [
    1,
    5,
    8,
    12,
    20,
    35,
    37,
    39,
    42,
    45
   ],
   [
    2,
    3,
    6,
    13,
    18,
    33,
    38,
    40,
    43,
    46
   ],
   [
    0,
    4,
    7,
    14,
    19,
    34,
    36,
    41,
    44,
    47
   ],
   [
    2,
    4,
    8,
    15,
    23,
    29,
    36,
    40,
    42,
    48
   ],
   [
    0,
    5,
    6,
    16,
    21,
    27,
    37,
    41,
    43,
    49
   ],
   [
    1,
    3,
    7,
    17,
    22,
    28,
    38,
    39,
    44,
    50
   ],
   [
    8,
    11,
    14,
    16,
    18,
    33,
    41,
    45,
    48,
    52
   ],
   [
    6,
    9,
    12,
    17,
    19,
    34,
    39,
    46,
    49,
    53
   ],
   [
    7,
    10,
    13,
    15,
    20,
    35,
    40,
    47,
    50,
    51
   ],
   [
    2,
    10,
    14,
    17,
    21,
    27,
    44,
    46,
    48,
    51
   ],
   [
    0,
    11,
    12,
    15,
    22,
    28,
    42,
    47,
    49,
    52
   ],
   [
    1,
    9,
    13,
    16,
    23,
    29,
    43,
    45,
    50,
    53
   ],
   [
    5,
    11,
    13,
    17,
    24,
    30,
    38,
    45,
    49,
    51
   ],
   [
    3,
    9,
    14,
    15,
    25,
    31,
    36,
    46,
    50,
    52
   ],
   [
    4,
    10,
    12,
    16,
    26,
    32,
    37,
    47,
    48,
    53
   ]
  ],
  "Z": [
   [
    0,
    5,
    6,
    16,
    21,
    27,
    37,
    41,
    43,
    49
   ],
   [
    1,
    3,
    7,
    17,
    22,
    28,
    38,
    39,
    44,
    50
   ],
   [
    2,
    4,
    8,
    15,
    23,
    29,
    36,
    40,
    42,
    48
   ],
   [
    0,
    3,
    8,
    10,
    24,
    30,
    37,
    40,
    44,
    52
   ],
   [
    1,
    4,
    6,
    11,
    25,
    31,
    38,
    41,
    42,
    53
   ],
   [
    2,
    5,
    7,
    9,
    26,
    32,
    36,
    39,
    43,
    51
   ],
   [
    2,
    3,
    6,
    13,
    18,
    33,
    38,
    40,
    43,
    46
   ],
   [
    0,
    4,
    7,
    14,
    19,
    34,
    36,
    41,
    44,
    47
   ],
   [
    1,
    5,
    8,
    12,
    20,
    35,
    37,
    39,
    42,
    45
   ],
   [
    3,
    9,
    14,
    15,
    25,
    31,
    36,
    46,
    50,
    52
   ],
   [
    4,
    10,
    12,
    16,
    26,
    32,
    37,
    47,
    48,
    53
   ],
   [
    5,
    11,
    13,
    17,
    24,
    30,
    38,
    45,
    49,
    51
   ],
   [
    6,
    9,
    12,
    17,
    19,
    34,
    39,
    46,
    49,
    53
   ],
   [
    7,
    10,
    13,
    15,
    20,
    35,
    40,
    47,
    50,
    51
   ],
   [
    8,
    11,
    14,
    16,
    18,
    33,
    41,
    45,
    48,
    52
   ],
   [
    0,
    11,
    12,
    15,
    22,
    28,
    42,
    47,
    49,
    52
   ],
   [
    1,
    9,
    13,
    16,
    23,
    29,
    43,
    45,
    50,
    53
   ],
   [
    2,
    10,
    14,
    17,
    21,
    27,
    44,
    46,
    48,
    51
   ],
   [
    7,
    12,
    18,
    23,
    24,
    28,
    32,
    34,
    40,
    45
   ],
   [
    8,
    13,
    19,
    21,
    25,
    29,
    30,
    35,
    41,
    46
   ],
   [
    6,
    14,
    20,
    22,
    26,
    27,
    31,
    33,
    39,
    47
   ],
   [
    1,
    15,
    18,
    21,
    26,
    28,
    31,
    35,
    43,
    48
   ],
   [
    2,
    16,
    19,
    22,
    24,
    29,
    32,
    33,
    44,
    49
   ],
   [
    0,
    17,
    20,
    23,
    25,
    27,
    30,
    34,
    42,
    50
   ],
   [
    4,
    9,
    20,
    21,
    24,
    29,
    31,
    34,
    37,
    51
   ],
   [
    5,
    10,
    18,
    22,
    25,
    27,
    32,
    35,
    38,
    52
   ],
   [
    3,
    11,
    19,
    23,
    26,
    28,
    30,
    33,
    36,
    53
   ]
  ]
 },
 "distance": {
  "d": 4,
  "X": {
   "value": 4,
   "confidence": "upper_bound",
   "witness": [
    4,
    15,
    31,
    47
   ]
  },
  "Z": {
   "value": 4,
   "confidence": "upper_bound",
   "witness": [
    4,
    15,
    31,
    47
   ]
  }
 },
 "provenance": {
  "authors": [
   "@FarLab"
  ],
  "construction": "2BGA (two-block group algebra) code on group ab(3,3,3) (|G|=27); H_X=[L(a)|R(b)], H_Z=[R(b)^T|L(a)^T] with a = [0, 10, 14, 16, 22], b = [0, 5, 6, 16, 21] as element indices of the Cayley table (identity = 0; table convention in the note)",
  "origin": "submission",
  "date": "2026-10-17",
  "model": "Claude Opus 5.5",
  "search_budget": {
   "candidates_screened": 9880000,
   "ris_trials_per_side": 30000000,
   "cpu_hours": 3,
   "tool": "research/kit 2BGA sweep + gf2_fast on Brev n2d-highcpu-64"
  }
 },
 "family": "bivariate-bicycle",
 "circuit": {
  "d_circ": {
   "Z": {
    "value": 2,
    "confidence": "upper_bound",
    "witness": [
     1679,
     1680
    ]
   },
   "X": {
    "value": 2,
    "confidence": "upper_bound",
    "witness": [
     1292,
     1293
    ]
   }
  },
  "rounds": 4,
  "stim_version": "1.16.0",
  "notes": "Generic sequential two-block extraction: one ancilla per check, X checks then Z checks each round with joint reset and measurement layers; CX layers are a proper edge coloring of each Tanner graph (12 X + 12 Z CX layers per round, coloring candidate 0); generated by qldpc submit (research/circuit_autogen.py, seed 0), the first of up to 6 candidates whose RIS screen at 2 rounds found nothing below d; witnesses from RIS on ker(H_dem) at twice the gate's trial budget, validated in GF(2)"
 }
}