{
 "schema_version": "0.2",
 "name": "[[128,8,9]] two-block group algebra CSS (arXiv:2306.16400)",
 "code_type": "CSS",
 "n": 128,
 "k": 8,
 "checks": {
  "X": [
   [
    0,
    12,
    64,
    73,
    110,
    118
   ],
   [
    1,
    7,
    64,
    65,
    79,
    123
   ],
   [
    0,
    2,
    66,
    85,
    94,
    106
   ],
   [
    3,
    37,
    67,
    79,
    112,
    127
   ],
   [
    4,
    32,
    65,
    68,
    89,
    126
   ],
   [
    5,
    33,
    69,
    76,
    109,
    115
   ],
   [
    6,
    34,
    70,
    87,
    94,
    97
   ],
   [
    7,
    27,
    70,
    71,
    76,
    95
   ],
   [
    8,
    22,
    72,
    81,
    89,
    118
   ],
   [
    9,
    23,
    68,
    73,
    100,
    127
   ],
   [
    10,
    24,
    74,
    78,
    85,
    86
   ],
   [
    11,
    25,
    66,
    70,
    75,
    107
   ],
   [
    3,
    12,
    76,
    87,
    104,
    125
   ],
   [
    4,
    13,
    75,
    77,
    86,
    105
   ],
   [
    5,
    14,
    64,
    78,
    93,
    127
   ],
   [
    6,
    15,
    69,
    73,
    79,
    122
   ],
   [
    16,
    55,
    80,
    91,
    100,
    123
   ],
   [
    2,
    17,
    81,
    90,
    101,
    122
   ],
   [
    14,
    18,
    66,
    82,
    121,
    125
   ],
   [
    19,
    52,
    83,
    88,
    95,
    96
   ],
   [
    20,
    53,
    71,
    75,
    84,
    116
   ],
   [
    21,
    54,
    85,
    93,
    98,
    124
   ],
   [
    1,
    22,
    66,
    86,
    104,
    114
   ],
   [
    23,
    48,
    84,
    87,
    96,
    115
   ],
   [
    24,
    28,
    65,
    69,
    88,
    118
   ],
   [
    25,
    50,
    64,
    74,
    89,
    98
   ],
   [
    26,
    42,
    90,
    102,
    110,
    126
   ],
   [
    27,
    43,
    67,
    91,
    98,
    111
   ],
   [
    28,
    44,
    71,
    92,
    97,
    104
   ],
   [
    29,
    45,
    93,
    99,
    105,
    106
   ],
   [
    30,
    46,
    77,
    84,
    94,
    121
   ],
   [
    31,
    47,
    69,
    95,
    101,
    108
   ],
   [
    16,
    32,
    71,
    96,
    114,
    120
   ],
   [
    17,
    33,
    67,
    97,
    113,
    124
   ],
   [
    18,
    34,
    81,
    98,
    110,
    112
   ],
   [
    19,
    35,
    68,
    74,
    99,
    123
   ],
   [
    20,
    36,
    65,
    83,
    100,
    108
   ],
   [
    21,
    37,
    70,
    101,
    115,
    121
   ],
   [
    13,
    38,
    72,
    102,
    108,
    119
   ],
   [
    35,
    39,
    77,
    103,
    107,
    114
   ],
   [
    15,
    40,
    78,
    104,
    106,
    113
   ],
   [
    41,
    62,
    74,
    105,
    111,
    117
   ],
   [
    9,
    42,
    77,
    106,
    120,
    125
   ],
   [
    10,
    43,
    72,
    82,
    101,
    107
   ],
   [
    11,
    44,
    79,
    81,
    91,
    108
   ],
   [
    45,
    49,
    73,
    83,
    109,
    126
   ],
   [
    46,
    61,
    68,
    93,
    110,
    117
   ],
   [
    8,
    47,
    75,
    85,
    97,
    111
   ],
   [
    48,
    57,
    80,
    112,
    117,
    124
   ],
   [
    49,
    58,
    87,
    113,
    116,
    120
   ],
   [
    50,
    59,
    76,
    82,
    88,
    114
   ],
   [
    51,
    60,
    83,
    115,
    119,
    122
   ],
   [
    38,
    52,
    80,
    92,
    111,
    116
   ],
   [
    39,
    53,
    89,
    91,
    102,
    117
   ],
   [
    40,
    54,
    82,
    90,
    109,
    118
   ],
   [
    41,
    55,
    84,
    95,
    107,
    119
   ],
   [
    36,
    56,
    86,
    92,
    99,
    120
   ],
   [
    29,
    57,
    90,
    103,
    119,
    121
   ],
   [
    30,
    58,
    100,
    102,
    112,
    122
   ],
   [
    31,
    59,
    67,
    78,
    92,
    123
   ],
   [
    26,
    60,
    94,
    105,
    116,
    124
   ],
   [
    61,
    63,
    96,
    103,
    109,
    125
   ],
   [
    56,
    62,
    72,
    88,
    103,
    126
   ],
   [
    51,
    63,
    80,
    99,
    113,
    127
   ]
  ],
  "Z": [
   [
    0,
    1,
    14,
    25,
    64,
    66
   ],
   [
    1,
    4,
    24,
    36,
    65,
    86
   ],
   [
    2,
    11,
    18,
    22,
    66,
    81
   ],
   [
    3,
    27,
    33,
    59,
    67,
    76
   ],
   [
    4,
    9,
    35,
    46,
    68,
    77
   ],
   [
    5,
    15,
    24,
    31,
    69,
    78
   ],
   [
    6,
    7,
    11,
    37,
    70,
    79
   ],
   [
    7,
    20,
    28,
    32,
    65,
    71
   ],
   [
    8,
    38,
    43,
    62,
    72,
    111
   ],
   [
    0,
    9,
    15,
    45,
    73,
    106
   ],
   [
    10,
    25,
    35,
    41,
    74,
    107
   ],
   [
    11,
    13,
    20,
    47,
    75,
    108
   ],
   [
    5,
    7,
    12,
    50,
    64,
    76
   ],
   [
    13,
    30,
    39,
    42,
    77,
    102
   ],
   [
    10,
    14,
    40,
    59,
    78,
    82
   ],
   [
    1,
    3,
    15,
    44,
    79,
    104
   ],
   [
    16,
    48,
    52,
    63,
    80,
    96
   ],
   [
    8,
    17,
    34,
    44,
    81,
    97
   ],
   [
    18,
    43,
    50,
    54,
    82,
    98
   ],
   [
    19,
    36,
    45,
    51,
    83,
    99
   ],
   [
    20,
    23,
    30,
    55,
    84,
    100
   ],
   [
    2,
    10,
    21,
    47,
    85,
    101
   ],
   [
    10,
    13,
    22,
    56,
    72,
    86
   ],
   [
    6,
    12,
    23,
    49,
    73,
    87
   ],
   [
    19,
    24,
    50,
    62,
    74,
    88
   ],
   [
    4,
    8,
    25,
    53,
    75,
    89
   ],
   [
    17,
    26,
    54,
    57,
    90,
    124
   ],
   [
    16,
    27,
    44,
    53,
    71,
    91
   ],
   [
    28,
    52,
    56,
    59,
    88,
    92
   ],
   [
    14,
    21,
    29,
    46,
    93,
    121
   ],
   [
    2,
    6,
    30,
    60,
    94,
    122
   ],
   [
    7,
    19,
    31,
    55,
    95,
    123
   ],
   [
    19,
    23,
    32,
    61,
    68,
    96
   ],
   [
    6,
    28,
    33,
    47,
    69,
    97
   ],
   [
    21,
    25,
    27,
    34,
    70,
    98
   ],
   [
    29,
    35,
    56,
    63,
    99,
    103
   ],
   [
    9,
    16,
    36,
    58,
    100,
    120
   ],
   [
    17,
    31,
    37,
    43,
    67,
    101
   ],
   [
    26,
    38,
    53,
    58,
    102,
    116
   ],
   [
    39,
    57,
    61,
    62,
    103,
    117
   ],
   [
    12,
    22,
    28,
    40,
    104,
    118
   ],
   [
    13,
    29,
    41,
    60,
    105,
    119
   ],
   [
    2,
    29,
    40,
    42,
    90,
    106
   ],
   [
    11,
    39,
    43,
    55,
    91,
    107
   ],
   [
    31,
    36,
    38,
    44,
    92,
    108
   ],
   [
    5,
    45,
    54,
    61,
    93,
    109
   ],
   [
    0,
    26,
    34,
    46,
    94,
    110
   ],
   [
    27,
    41,
    47,
    52,
    95,
    111
   ],
   [
    3,
    34,
    48,
    58,
    87,
    112
   ],
   [
    33,
    40,
    49,
    63,
    109,
    113
   ],
   [
    22,
    32,
    39,
    50,
    89,
    114
   ],
   [
    5,
    23,
    37,
    51,
    115,
    127
   ],
   [
    20,
    49,
    52,
    60,
    83,
    116
   ],
   [
    41,
    46,
    48,
    53,
    84,
    117
   ],
   [
    0,
    8,
    24,
    54,
    85,
    118
   ],
   [
    38,
    51,
    55,
    57,
    80,
    119
   ],
   [
    32,
    42,
    49,
    56,
    120,
    126
   ],
   [
    18,
    30,
    37,
    57,
    112,
    121
   ],
   [
    15,
    17,
    51,
    58,
    113,
    122
   ],
   [
    1,
    16,
    35,
    59,
    114,
    123
   ],
   [
    21,
    33,
    48,
    60,
    115,
    124
   ],
   [
    12,
    18,
    42,
    61,
    110,
    125
   ],
   [
    4,
    26,
    45,
    62,
    105,
    126
   ],
   [
    3,
    9,
    14,
    63,
    125,
    127
   ]
  ]
 },
 "distance": {
  "d": 9,
  "X": {
   "value": 9,
   "confidence": "upper_bound",
   "witness": [
    0,
    14,
    54,
    82,
    90,
    93,
    106,
    110,
    125
   ],
   "witness_provenance": {
    "found_by": [
     "@vprusso"
    ],
    "date": "2026-10-17",
    "found_at_samples": 40000,
    "survived_samples": 40000,
    "tool": "research/kit/surrogate.lightest_logical",
    "seeds": [
     20261017
    ]
   }
  },
  "Z": {
   "value": 9,
   "confidence": "upper_bound",
   "witness": [
    30,
    38,
    48,
    57,
    58,
    60,
    84,
    116,
    119
   ],
   "witness_provenance": {
    "found_by": [
     "@vprusso"
    ],
    "date": "2026-10-17",
    "found_at_samples": 40000,
    "survived_samples": 40000,
    "tool": "research/kit/surrogate.lightest_logical",
    "seeds": [
     20261018
    ]
   }
  }
 },
 "provenance": {
  "authors": [
   "Lin, Hsiang-Ku",
   "Pryadko, Leonid P."
  ],
  "model": "Claude Fable 5.1 (Claude Code)",
  "construction": "Two-block group-algebra (2BGA) code over SmallGroup(64,47) = C16 : C4, archive row \"64 47 8 9 [ 3 ] [ 2, 15, 26 ]\" of the nonabelian enumeration (Lin and Pryadko, arXiv:2306.16400; github.com/QEC-pages/2BGA-codes @ 403d194c3f98). a = {g_i : i in [3]} and b = {g_i : i in [2, 15, 26]} with elements indexed as GAP's Elements(SmallGroup(64,47)) numbers them, 1-based (the identity is implicit in the archive's lists); H_X = [L(a) | R(b)], H_Z = [R(b)^T | L(a)^T] over the left and right regular representations. Rebuilt here as research/kit/group_algebra.build_2bga_small_group(64, 47, [0, 2], [0, 1, 14, 25]) on the committed Cayley table research/kit/groups/smallgroup_64_47.json (GAP 4.16.1, smallgrp 1.7.0), 0-based indices with the identity at 0.",
  "references": [
   "arXiv:2306.16400",
   "https://github.com/QEC-pages/2BGA-codes"
  ],
  "date": "2023",
  "notes": "Literature baseline (arXiv:2306.16400, Lin and Pryadko, 28 Jun 2023), seeded for issue #3658 as the 2BGA archive row that dominates 128-8-6 at equal n, k, and check weight. Reconstructed here from the archive row with the group table exported by research/kit/groups/cayley_table.g; n, k, and the check weight match the row, and the reconstruction convention was checked by rebuilding five archive rows already on the board and finding each permutation-equivalent to its entry. The archive's d = 9 is a QDistRnd upper bound; this entry claims d <= 9 from its own witnesses (40,000 RIS trials per side on the NumPy backend, seed 20261017, X 9, Z 9), and 2 x 2,000,000 gf2_fast trials found nothing lighter than 9.",
  "origin": "baseline",
  "novelty": "known_parameters"
 },
 "family": "2bga-nonabelian"
}