{
 "schema_version": "0.2",
 "name": "[[168,16,16]] two-block code over C84",
 "code_type": "CSS",
 "n": 168,
 "k": 16,
 "checks": {
  "X": [
   [
    0,
    19,
    27,
    51,
    84,
    109,
    123,
    162
   ],
   [
    1,
    9,
    25,
    32,
    85,
    92,
    104,
    165
   ],
   [
    2,
    4,
    37,
    61,
    86,
    91,
    133,
    166
   ],
   [
    3,
    31,
    39,
    63,
    87,
    94,
    121,
    135
   ],
   [
    4,
    8,
    14,
    38,
    88,
    97,
    110,
    152
   ],
   [
    5,
    7,
    18,
    42,
    84,
    89,
    114,
    167
   ],
   [
    6,
    20,
    37,
    44,
    90,
    99,
    102,
    116
   ],
   [
    7,
    13,
    21,
    45,
    91,
    103,
    117,
    157
   ],
   [
    8,
    10,
    17,
    41,
    92,
    99,
    113,
    154
   ],
   [
    9,
    12,
    49,
    72,
    93,
    101,
    103,
    145
   ],
   [
    0,
    10,
    24,
    48,
    85,
    94,
    120,
    160
   ],
   [
    11,
    43,
    51,
    74,
    95,
    105,
    133,
    147
   ],
   [
    12,
    18,
    26,
    50,
    86,
    96,
    108,
    122
   ],
   [
    3,
    13,
    15,
    22,
    86,
    95,
    97,
    159
   ],
   [
    14,
    17,
    30,
    54,
    87,
    98,
    109,
    126
   ],
   [
    1,
    15,
    31,
    55,
    88,
    99,
    127,
    163
   ],
   [
    16,
    32,
    49,
    56,
    100,
    111,
    114,
    128
   ],
   [
    17,
    24,
    33,
    57,
    89,
    101,
    115,
    129
   ],
   [
    18,
    21,
    29,
    53,
    88,
    102,
    111,
    125
   ],
   [
    2,
    6,
    19,
    28,
    89,
    100,
    103,
    142
   ],
   [
    20,
    23,
    61,
    80,
    104,
    113,
    115,
    156
   ],
   [
    3,
    21,
    36,
    60,
    90,
    92,
    105,
    132
   ],
   [
    22,
    55,
    63,
    81,
    106,
    117,
    145,
    158
   ],
   [
    23,
    30,
    38,
    62,
    93,
    107,
    120,
    134
   ],
   [
    11,
    24,
    27,
    34,
    91,
    93,
    106,
    108
   ],
   [
    5,
    12,
    25,
    35,
    94,
    107,
    109,
    148
   ],
   [
    26,
    29,
    42,
    66,
    95,
    110,
    121,
    138
   ],
   [
    6,
    27,
    43,
    67,
    96,
    97,
    111,
    139
   ],
   [
    28,
    44,
    61,
    68,
    112,
    123,
    126,
    140
   ],
   [
    29,
    36,
    45,
    69,
    98,
    113,
    127,
    141
   ],
   [
    30,
    33,
    41,
    65,
    96,
    114,
    123,
    137
   ],
   [
    9,
    16,
    31,
    40,
    84,
    98,
    112,
    115
   ],
   [
    32,
    35,
    72,
    83,
    116,
    125,
    127,
    164
   ],
   [
    11,
    33,
    48,
    71,
    100,
    102,
    117,
    144
   ],
   [
    15,
    34,
    67,
    74,
    118,
    129,
    156,
    165
   ],
   [
    35,
    42,
    50,
    73,
    104,
    119,
    132,
    146
   ],
   [
    22,
    36,
    39,
    46,
    101,
    104,
    118,
    120
   ],
   [
    14,
    23,
    37,
    47,
    85,
    105,
    119,
    121
   ],
   [
    38,
    41,
    54,
    76,
    106,
    122,
    133,
    150
   ],
   [
    16,
    39,
    55,
    77,
    107,
    108,
    123,
    151
   ],
   [
    2,
    40,
    56,
    72,
    124,
    135,
    138,
    152
   ],
   [
    41,
    48,
    57,
    78,
    110,
    125,
    139,
    153
   ],
   [
    42,
    45,
    53,
    75,
    107,
    126,
    135,
    149
   ],
   [
    20,
    28,
    43,
    52,
    87,
    110,
    124,
    127
   ],
   [
    25,
    44,
    47,
    80,
    128,
    137,
    139,
    167
   ],
   [
    22,
    45,
    60,
    79,
    112,
    114,
    129,
    155
   ],
   [
    27,
    46,
    77,
    81,
    99,
    130,
    141,
    164
   ],
   [
    5,
    47,
    54,
    62,
    116,
    131,
    144,
    157
   ],
   [
    34,
    48,
    51,
    58,
    113,
    116,
    130,
    132
   ],
   [
    26,
    35,
    49,
    59,
    90,
    117,
    131,
    133
   ],
   [
    8,
    50,
    53,
    66,
    118,
    134,
    145,
    160
   ],
   [
    28,
    51,
    67,
    82,
    119,
    120,
    135,
    161
   ],
   [
    9,
    52,
    68,
    80,
    86,
    136,
    147,
    150
   ],
   [
    10,
    53,
    60,
    69,
    122,
    137,
    151,
    162
   ],
   [
    7,
    54,
    57,
    65,
    119,
    138,
    147,
    159
   ],
   [
    32,
    40,
    55,
    64,
    95,
    122,
    136,
    139
   ],
   [
    37,
    56,
    59,
    83,
    109,
    140,
    149,
    151
   ],
   [
    13,
    34,
    57,
    71,
    124,
    126,
    141,
    163
   ],
   [
    15,
    39,
    58,
    82,
    111,
    142,
    153,
    167
   ],
   [
    14,
    59,
    66,
    73,
    89,
    128,
    143,
    155
   ],
   [
    0,
    46,
    60,
    63,
    125,
    128,
    142,
    144
   ],
   [
    38,
    47,
    61,
    70,
    100,
    129,
    143,
    145
   ],
   [
    18,
    62,
    65,
    76,
    92,
    130,
    146,
    156
   ],
   [
    19,
    40,
    63,
    77,
    131,
    132,
    147,
    166
   ],
   [
    2,
    20,
    64,
    83,
    93,
    148,
    158,
    160
   ],
   [
    21,
    65,
    71,
    78,
    94,
    134,
    149,
    161
   ],
   [
    17,
    66,
    69,
    75,
    91,
    131,
    150,
    158
   ],
   [
    1,
    44,
    52,
    67,
    106,
    134,
    148,
    151
   ],
   [
    25,
    49,
    68,
    70,
    121,
    152,
    159,
    161
   ],
   [
    24,
    46,
    69,
    79,
    97,
    136,
    138,
    153
   ],
   [
    5,
    26,
    70,
    76,
    98,
    140,
    154,
    163
   ],
   [
    3,
    58,
    71,
    74,
    84,
    137,
    140,
    155
   ],
   [
    4,
    50,
    59,
    72,
    112,
    141,
    154,
    156
   ],
   [
    8,
    30,
    73,
    75,
    102,
    142,
    157,
    164
   ],
   [
    31,
    52,
    74,
    82,
    103,
    143,
    144,
    158
   ],
   [
    10,
    33,
    75,
    79,
    105,
    146,
    159,
    166
   ],
   [
    7,
    29,
    76,
    78,
    101,
    143,
    160,
    165
   ],
   [
    6,
    56,
    64,
    77,
    85,
    118,
    146,
    161
   ],
   [
    13,
    36,
    58,
    78,
    108,
    148,
    150,
    162
   ],
   [
    0,
    11,
    79,
    81,
    87,
    149,
    152,
    163
   ],
   [
    12,
    62,
    70,
    80,
    88,
    124,
    153,
    164
   ],
   [
    19,
    43,
    64,
    81,
    115,
    154,
    155,
    165
   ],
   [
    1,
    16,
    68,
    82,
    90,
    130,
    157,
    166
   ],
   [
    4,
    23,
    73,
    83,
    96,
    136,
    162,
    167
   ]
  ],
  "Z": [
   [
    0,
    5,
    31,
    71,
    84,
    94,
    144,
    163
   ],
   [
    1,
    10,
    37,
    77,
    85,
    99,
    151,
    166
   ],
   [
    2,
    12,
    13,
    52,
    86,
    103,
    124,
    148
   ],
   [
    3,
    14,
    43,
    79,
    87,
    97,
    105,
    155
   ],
   [
    4,
    15,
    18,
    80,
    86,
    88,
    156,
    167
   ],
   [
    5,
    17,
    19,
    59,
    89,
    109,
    131,
    154
   ],
   [
    6,
    21,
    49,
    82,
    90,
    103,
    111,
    161
   ],
   [
    2,
    7,
    24,
    66,
    89,
    91,
    138,
    160
   ],
   [
    1,
    8,
    21,
    62,
    88,
    92,
    134,
    157
   ],
   [
    9,
    23,
    24,
    64,
    85,
    93,
    115,
    136
   ],
   [
    3,
    10,
    25,
    65,
    92,
    94,
    137,
    159
   ],
   [
    11,
    13,
    26,
    55,
    95,
    108,
    117,
    163
   ],
   [
    12,
    27,
    30,
    83,
    93,
    96,
    109,
    164
   ],
   [
    4,
    13,
    27,
    69,
    91,
    97,
    141,
    162
   ],
   [
    14,
    29,
    31,
    70,
    88,
    98,
    121,
    143
   ],
   [
    6,
    8,
    15,
    46,
    97,
    99,
    118,
    142
   ],
   [
    16,
    19,
    33,
    61,
    100,
    115,
    123,
    166
   ],
   [
    9,
    17,
    36,
    76,
    92,
    98,
    101,
    150
   ],
   [
    6,
    18,
    33,
    73,
    89,
    96,
    102,
    146
   ],
   [
    7,
    9,
    19,
    74,
    84,
    103,
    147,
    165
   ],
   [
    1,
    20,
    35,
    36,
    90,
    104,
    127,
    148
   ],
   [
    11,
    21,
    37,
    75,
    91,
    102,
    105,
    149
   ],
   [
    22,
    24,
    38,
    67,
    97,
    106,
    120,
    129
   ],
   [
    23,
    25,
    39,
    42,
    104,
    107,
    121,
    167
   ],
   [
    12,
    24,
    39,
    78,
    94,
    101,
    108,
    153
   ],
   [
    0,
    14,
    25,
    56,
    85,
    109,
    128,
    152
   ],
   [
    4,
    26,
    41,
    43,
    96,
    110,
    133,
    154
   ],
   [
    16,
    18,
    27,
    58,
    84,
    108,
    111,
    130
   ],
   [
    28,
    31,
    45,
    72,
    103,
    112,
    127,
    135
   ],
   [
    8,
    20,
    29,
    48,
    102,
    110,
    113,
    160
   ],
   [
    5,
    16,
    30,
    45,
    98,
    107,
    114,
    157
   ],
   [
    17,
    20,
    31,
    81,
    87,
    99,
    115,
    158
   ],
   [
    6,
    32,
    47,
    48,
    85,
    100,
    116,
    139
   ],
   [
    7,
    22,
    33,
    49,
    101,
    114,
    117,
    159
   ],
   [
    34,
    36,
    50,
    77,
    108,
    118,
    132,
    141
   ],
   [
    35,
    37,
    51,
    54,
    109,
    116,
    119,
    133
   ],
   [
    10,
    23,
    36,
    51,
    105,
    113,
    120,
    162
   ],
   [
    3,
    26,
    37,
    68,
    86,
    90,
    121,
    140
   ],
   [
    12,
    38,
    53,
    55,
    88,
    107,
    122,
    145
   ],
   [
    0,
    28,
    30,
    39,
    87,
    120,
    123,
    142
   ],
   [
    40,
    43,
    57,
    80,
    115,
    124,
    139,
    147
   ],
   [
    18,
    32,
    41,
    60,
    92,
    114,
    122,
    125
   ],
   [
    14,
    28,
    42,
    57,
    89,
    110,
    119,
    126
   ],
   [
    15,
    29,
    32,
    43,
    95,
    111,
    127,
    165
   ],
   [
    16,
    44,
    59,
    60,
    90,
    112,
    128,
    151
   ],
   [
    17,
    34,
    45,
    61,
    91,
    113,
    126,
    129
   ],
   [
    46,
    48,
    62,
    82,
    120,
    130,
    144,
    153
   ],
   [
    47,
    49,
    63,
    66,
    121,
    128,
    131,
    145
   ],
   [
    21,
    35,
    48,
    63,
    94,
    117,
    125,
    132
   ],
   [
    2,
    11,
    38,
    49,
    93,
    100,
    133,
    152
   ],
   [
    23,
    50,
    65,
    67,
    96,
    119,
    134,
    156
   ],
   [
    3,
    40,
    42,
    51,
    84,
    95,
    132,
    135
   ],
   [
    52,
    55,
    69,
    83,
    127,
    136,
    151,
    158
   ],
   [
    30,
    44,
    53,
    71,
    102,
    126,
    134,
    137
   ],
   [
    26,
    40,
    54,
    69,
    98,
    122,
    131,
    138
   ],
   [
    27,
    41,
    44,
    55,
    99,
    106,
    123,
    139
   ],
   [
    28,
    56,
    70,
    71,
    100,
    124,
    140,
    161
   ],
   [
    29,
    46,
    57,
    72,
    101,
    125,
    138,
    141
   ],
   [
    19,
    58,
    60,
    73,
    132,
    142,
    155,
    162
   ],
   [
    59,
    61,
    74,
    76,
    133,
    140,
    143,
    156
   ],
   [
    33,
    47,
    60,
    74,
    105,
    129,
    137,
    144
   ],
   [
    9,
    22,
    50,
    61,
    86,
    104,
    112,
    145
   ],
   [
    35,
    62,
    75,
    77,
    107,
    131,
    146,
    164
   ],
   [
    11,
    52,
    54,
    63,
    87,
    106,
    144,
    147
   ],
   [
    25,
    64,
    67,
    78,
    139,
    148,
    161,
    165
   ],
   [
    42,
    56,
    65,
    79,
    114,
    138,
    146,
    149
   ],
   [
    38,
    52,
    66,
    78,
    110,
    134,
    143,
    150
   ],
   [
    39,
    53,
    56,
    67,
    111,
    118,
    135,
    151
   ],
   [
    4,
    40,
    68,
    79,
    112,
    136,
    152,
    166
   ],
   [
    41,
    58,
    69,
    80,
    113,
    137,
    150,
    153
   ],
   [
    8,
    70,
    72,
    81,
    145,
    152,
    154,
    164
   ],
   [
    45,
    59,
    71,
    81,
    117,
    141,
    149,
    155
   ],
   [
    20,
    34,
    62,
    72,
    93,
    116,
    124,
    156
   ],
   [
    7,
    47,
    73,
    82,
    119,
    143,
    157,
    167
   ],
   [
    22,
    64,
    66,
    74,
    95,
    118,
    155,
    158
   ],
   [
    13,
    54,
    68,
    75,
    126,
    150,
    157,
    159
   ],
   [
    10,
    50,
    64,
    76,
    122,
    146,
    154,
    160
   ],
   [
    51,
    65,
    68,
    77,
    123,
    130,
    147,
    161
   ],
   [
    0,
    53,
    78,
    83,
    125,
    149,
    160,
    162
   ],
   [
    15,
    57,
    70,
    79,
    129,
    153,
    159,
    163
   ],
   [
    32,
    46,
    73,
    80,
    104,
    128,
    136,
    164
   ],
   [
    1,
    34,
    76,
    81,
    106,
    130,
    163,
    165
   ],
   [
    2,
    63,
    75,
    82,
    135,
    142,
    158,
    166
   ],
   [
    5,
    44,
    58,
    83,
    116,
    140,
    148,
    167
   ]
  ]
 },
 "distance": {
  "d": 16,
  "X": {
   "value": 16,
   "confidence": "upper_bound",
   "witness": [
    1,
    10,
    12,
    13,
    35,
    46,
    54,
    81,
    87,
    92,
    97,
    101,
    117,
    120,
    122,
    164
   ],
   "witness_provenance": {
    "found_by": [
     "@mathysrennela"
    ],
    "date": "2026-10-12",
    "found_at_samples": 2000,
    "survived_samples": 2000,
    "tool": "research/kit/surrogate.lightest_logical",
    "seeds": [
     1011
    ]
   }
  },
  "Z": {
   "value": 16,
   "confidence": "upper_bound",
   "witness": [
    11,
    20,
    33,
    38,
    41,
    57,
    61,
    68,
    72,
    93,
    99,
    127,
    141,
    147,
    152,
    166
   ],
   "witness_provenance": {
    "found_by": [
     "@mathysrennela"
    ],
    "date": "2026-10-12",
    "found_at_samples": 2000,
    "survived_samples": 2000,
    "tool": "research/kit/surrogate.lightest_logical",
    "seeds": [
     1012
    ]
   }
  }
 },
 "provenance": {
  "authors": [
   "Lee, Gideon"
  ],
  "construction": "Two-block group-algebra code C(a,b) over C84 = SmallGroup(84, 6) (order 84), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 168; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 10, 60, 79}, b = {0, 5, 31, 71} as element indices. Check weight 8. Parameters and blocks taken from the public code database of arXiv:2610.13880v1 (Lee, 'Bicycle Race'), github.com/gideonlsx/bicycle-race.",
  "references": [
   "arXiv:2610.13880"
  ],
  "date": "2026-10-12",
  "notes": "Parameters and construction taken from the public database of arXiv:2610.13880v1. Distance is an upper bound certified by the witnesses attached here. Parameters and blocks taken from the public code database of arXiv:2610.13880v1 (Lee, 'Bicycle Race'), github.com/gideonlsx/bicycle-race @ aed705a, CC BY 4.0; rebuilt here with the repo's own group data. Distance is an upper bound certified by the attached witnesses, not the paper's exact distance.",
  "origin": "baseline",
  "novelty": "known_parameters",
  "model": "Mimo-V2.6-Flash"
 },
 "family": "generalized-bicycle"
}