{
 "schema_version": "0.2",
 "name": "[[234,8,21]] two-block code over C3 x (C13 : C3)",
 "code_type": "CSS",
 "n": 234,
 "k": 8,
 "checks": {
  "X": [
   [
    0,
    4,
    102,
    105,
    117,
    150,
    223
   ],
   [
    0,
    1,
    54,
    75,
    118,
    156,
    228
   ],
   [
    2,
    10,
    86,
    89,
    119,
    134,
    229
   ],
   [
    3,
    11,
    110,
    112,
    119,
    120,
    159
   ],
   [
    1,
    4,
    90,
    112,
    121,
    162,
    231
   ],
   [
    2,
    5,
    38,
    59,
    122,
    140,
    232
   ],
   [
    3,
    6,
    63,
    84,
    122,
    123,
    165
   ],
   [
    7,
    18,
    94,
    97,
    124,
    142,
    215
   ],
   [
    8,
    19,
    95,
    98,
    124,
    125,
    143
   ],
   [
    7,
    9,
    20,
    115,
    125,
    126,
    168
   ],
   [
    5,
    10,
    74,
    98,
    127,
    146,
    233
   ],
   [
    6,
    7,
    11,
    99,
    127,
    128,
    171
   ],
   [
    7,
    12,
    46,
    67,
    129,
    148,
    221
   ],
   [
    8,
    13,
    47,
    68,
    129,
    130,
    149
   ],
   [
    9,
    14,
    72,
    93,
    130,
    131,
    174
   ],
   [
    15,
    27,
    103,
    106,
    117,
    132,
    151
   ],
   [
    0,
    16,
    28,
    104,
    132,
    133,
    152
   ],
   [
    12,
    15,
    17,
    29,
    133,
    134,
    177
   ],
   [
    12,
    18,
    82,
    106,
    135,
    154,
    226
   ],
   [
    0,
    13,
    19,
    83,
    135,
    136,
    155
   ],
   [
    14,
    15,
    20,
    107,
    136,
    137,
    180
   ],
   [
    15,
    21,
    55,
    76,
    118,
    138,
    157
   ],
   [
    16,
    22,
    56,
    77,
    138,
    139,
    158
   ],
   [
    17,
    23,
    81,
    102,
    139,
    140,
    183
   ],
   [
    2,
    24,
    36,
    111,
    120,
    141,
    160
   ],
   [
    1,
    3,
    25,
    37,
    141,
    142,
    161
   ],
   [
    21,
    24,
    26,
    38,
    142,
    143,
    186
   ],
   [
    2,
    21,
    27,
    91,
    121,
    144,
    163
   ],
   [
    3,
    22,
    28,
    92,
    144,
    145,
    164
   ],
   [
    23,
    24,
    29,
    113,
    145,
    146,
    189
   ],
   [
    24,
    30,
    64,
    85,
    123,
    147,
    166
   ],
   [
    25,
    31,
    65,
    86,
    147,
    148,
    167
   ],
   [
    26,
    32,
    90,
    110,
    148,
    149,
    192
   ],
   [
    5,
    8,
    33,
    45,
    126,
    150,
    169
   ],
   [
    6,
    9,
    34,
    46,
    150,
    151,
    170
   ],
   [
    30,
    33,
    35,
    47,
    151,
    152,
    195
   ],
   [
    8,
    30,
    36,
    100,
    128,
    153,
    172
   ],
   [
    9,
    31,
    37,
    101,
    153,
    154,
    173
   ],
   [
    32,
    33,
    38,
    116,
    154,
    155,
    198
   ],
   [
    33,
    39,
    73,
    94,
    131,
    156,
    175
   ],
   [
    34,
    40,
    74,
    95,
    156,
    157,
    176
   ],
   [
    35,
    41,
    99,
    115,
    157,
    158,
    201
   ],
   [
    13,
    16,
    42,
    54,
    134,
    159,
    178
   ],
   [
    14,
    17,
    43,
    55,
    159,
    160,
    179
   ],
   [
    39,
    42,
    44,
    56,
    160,
    161,
    204
   ],
   [
    16,
    39,
    45,
    108,
    137,
    162,
    181
   ],
   [
    17,
    40,
    46,
    109,
    162,
    163,
    182
   ],
   [
    18,
    41,
    42,
    47,
    163,
    164,
    207
   ],
   [
    42,
    48,
    82,
    103,
    140,
    165,
    184
   ],
   [
    43,
    49,
    83,
    104,
    165,
    166,
    185
   ],
   [
    12,
    44,
    50,
    107,
    166,
    167,
    210
   ],
   [
    22,
    25,
    51,
    63,
    143,
    168,
    187
   ],
   [
    23,
    26,
    52,
    64,
    168,
    169,
    188
   ],
   [
    48,
    51,
    53,
    65,
    169,
    170,
    213
   ],
   [
    25,
    48,
    54,
    114,
    146,
    171,
    190
   ],
   [
    4,
    26,
    49,
    55,
    171,
    172,
    191
   ],
   [
    27,
    50,
    51,
    56,
    172,
    173,
    216
   ],
   [
    51,
    57,
    91,
    111,
    149,
    174,
    193
   ],
   [
    1,
    52,
    58,
    92,
    174,
    175,
    194
   ],
   [
    21,
    53,
    59,
    113,
    175,
    176,
    219
   ],
   [
    31,
    34,
    60,
    72,
    152,
    177,
    196
   ],
   [
    32,
    35,
    61,
    73,
    177,
    178,
    197
   ],
   [
    57,
    60,
    62,
    74,
    178,
    179,
    222
   ],
   [
    10,
    34,
    57,
    63,
    155,
    180,
    199
   ],
   [
    11,
    35,
    58,
    64,
    180,
    181,
    200
   ],
   [
    36,
    59,
    60,
    65,
    181,
    182,
    224
   ],
   [
    5,
    60,
    66,
    100,
    158,
    183,
    202
   ],
   [
    6,
    61,
    67,
    101,
    183,
    184,
    203
   ],
   [
    30,
    62,
    68,
    116,
    184,
    185,
    227
   ],
   [
    40,
    43,
    69,
    81,
    161,
    186,
    205
   ],
   [
    41,
    44,
    70,
    82,
    186,
    187,
    206
   ],
   [
    66,
    69,
    71,
    83,
    187,
    188,
    229
   ],
   [
    19,
    43,
    66,
    72,
    164,
    189,
    208
   ],
   [
    20,
    44,
    67,
    73,
    189,
    190,
    209
   ],
   [
    45,
    68,
    69,
    74,
    190,
    191,
    230
   ],
   [
    13,
    69,
    75,
    108,
    167,
    192,
    211
   ],
   [
    14,
    70,
    76,
    109,
    192,
    193,
    212
   ],
   [
    18,
    39,
    71,
    77,
    193,
    194,
    232
   ],
   [
    49,
    52,
    78,
    90,
    170,
    195,
    214
   ],
   [
    50,
    53,
    79,
    91,
    195,
    196,
    215
   ],
   [
    75,
    78,
    80,
    92,
    124,
    196,
    197
   ],
   [
    28,
    52,
    75,
    81,
    173,
    198,
    217
   ],
   [
    29,
    53,
    76,
    82,
    198,
    199,
    218
   ],
   [
    54,
    77,
    78,
    83,
    199,
    200,
    233
   ],
   [
    22,
    78,
    84,
    114,
    176,
    201,
    220
   ],
   [
    4,
    23,
    79,
    85,
    201,
    202,
    221
   ],
   [
    27,
    48,
    80,
    86,
    129,
    202,
    203
   ],
   [
    58,
    61,
    87,
    99,
    179,
    204,
    223
   ],
   [
    59,
    62,
    88,
    100,
    117,
    204,
    205
   ],
   [
    84,
    87,
    89,
    101,
    132,
    205,
    206
   ],
   [
    37,
    61,
    84,
    90,
    182,
    207,
    225
   ],
   [
    38,
    62,
    85,
    91,
    207,
    208,
    226
   ],
   [
    63,
    86,
    87,
    92,
    135,
    208,
    209
   ],
   [
    10,
    31,
    87,
    93,
    185,
    210,
    228
   ],
   [
    11,
    32,
    88,
    94,
    118,
    210,
    211
   ],
   [
    36,
    57,
    89,
    95,
    138,
    211,
    212
   ],
   [
    67,
    70,
    96,
    107,
    119,
    188,
    213
   ],
   [
    68,
    71,
    97,
    108,
    120,
    213,
    214
   ],
   [
    93,
    96,
    98,
    109,
    141,
    214,
    215
   ],
   [
    46,
    70,
    93,
    99,
    191,
    216,
    231
   ],
   [
    47,
    71,
    94,
    100,
    121,
    216,
    217
   ],
   [
    72,
    95,
    96,
    101,
    144,
    217,
    218
   ],
   [
    19,
    40,
    96,
    102,
    122,
    194,
    219
   ],
   [
    20,
    41,
    97,
    103,
    123,
    219,
    220
   ],
   [
    45,
    66,
    98,
    104,
    147,
    220,
    221
   ],
   [
    76,
    79,
    105,
    113,
    125,
    197,
    222
   ],
   [
    77,
    80,
    106,
    114,
    126,
    222,
    223
   ],
   [
    55,
    79,
    102,
    107,
    127,
    200,
    224
   ],
   [
    56,
    80,
    103,
    108,
    128,
    224,
    225
   ],
   [
    81,
    104,
    105,
    109,
    153,
    225,
    226
   ],
   [
    28,
    49,
    105,
    110,
    130,
    203,
    227
   ],
   [
    29,
    50,
    106,
    111,
    131,
    227,
    228
   ],
   [
    85,
    88,
    112,
    116,
    133,
    206,
    229
   ],
   [
    64,
    88,
    110,
    113,
    136,
    209,
    230
   ],
   [
    65,
    89,
    111,
    114,
    137,
    230,
    231
   ],
   [
    37,
    58,
    112,
    115,
    139,
    212,
    232
   ],
   [
    73,
    97,
    115,
    116,
    145,
    218,
    233
   ]
  ],
  "Z": [
   [
    0,
    15,
    88,
    117,
    118,
    133,
    136
   ],
   [
    1,
    21,
    94,
    118,
    121,
    142,
    175
   ],
   [
    2,
    3,
    96,
    119,
    122,
    141,
    144
   ],
   [
    3,
    24,
    97,
    120,
    123,
    142,
    145
   ],
   [
    4,
    27,
    100,
    117,
    121,
    172,
    202
   ],
   [
    5,
    6,
    102,
    122,
    127,
    150,
    183
   ],
   [
    6,
    30,
    103,
    123,
    128,
    151,
    184
   ],
   [
    7,
    8,
    80,
    124,
    126,
    128,
    129
   ],
   [
    8,
    9,
    105,
    125,
    130,
    150,
    153
   ],
   [
    9,
    33,
    106,
    126,
    131,
    151,
    154
   ],
   [
    10,
    11,
    107,
    119,
    127,
    180,
    210
   ],
   [
    11,
    36,
    108,
    120,
    128,
    181,
    211
   ],
   [
    12,
    13,
    86,
    129,
    134,
    135,
    167
   ],
   [
    13,
    14,
    110,
    130,
    136,
    159,
    192
   ],
   [
    14,
    39,
    111,
    131,
    137,
    160,
    193
   ],
   [
    15,
    16,
    89,
    132,
    134,
    137,
    138
   ],
   [
    16,
    17,
    112,
    133,
    139,
    159,
    162
   ],
   [
    2,
    17,
    42,
    134,
    140,
    160,
    163
   ],
   [
    18,
    19,
    92,
    124,
    135,
    164,
    194
   ],
   [
    19,
    20,
    113,
    125,
    136,
    189,
    219
   ],
   [
    20,
    45,
    114,
    126,
    137,
    190,
    220
   ],
   [
    21,
    22,
    95,
    138,
    143,
    144,
    176
   ],
   [
    22,
    23,
    115,
    139,
    145,
    168,
    201
   ],
   [
    5,
    23,
    48,
    140,
    146,
    169,
    202
   ],
   [
    24,
    25,
    98,
    141,
    143,
    146,
    147
   ],
   [
    7,
    25,
    26,
    142,
    148,
    168,
    171
   ],
   [
    8,
    26,
    51,
    143,
    149,
    169,
    172
   ],
   [
    27,
    28,
    101,
    132,
    144,
    173,
    203
   ],
   [
    28,
    29,
    116,
    133,
    145,
    198,
    227
   ],
   [
    10,
    29,
    54,
    134,
    146,
    199,
    228
   ],
   [
    30,
    31,
    104,
    147,
    152,
    153,
    185
   ],
   [
    12,
    31,
    32,
    148,
    154,
    177,
    210
   ],
   [
    13,
    32,
    57,
    149,
    155,
    178,
    211
   ],
   [
    0,
    33,
    34,
    150,
    152,
    155,
    156
   ],
   [
    15,
    34,
    35,
    151,
    157,
    177,
    180
   ],
   [
    16,
    35,
    60,
    152,
    158,
    178,
    181
   ],
   [
    36,
    37,
    109,
    141,
    153,
    182,
    212
   ],
   [
    18,
    37,
    38,
    142,
    154,
    207,
    232
   ],
   [
    19,
    38,
    63,
    122,
    143,
    155,
    208
   ],
   [
    1,
    39,
    40,
    156,
    161,
    162,
    194
   ],
   [
    21,
    40,
    41,
    157,
    163,
    186,
    219
   ],
   [
    22,
    41,
    66,
    158,
    164,
    187,
    220
   ],
   [
    3,
    42,
    43,
    159,
    161,
    164,
    165
   ],
   [
    24,
    43,
    44,
    160,
    166,
    186,
    189
   ],
   [
    25,
    44,
    69,
    161,
    167,
    187,
    190
   ],
   [
    4,
    45,
    46,
    150,
    162,
    191,
    221
   ],
   [
    27,
    46,
    47,
    129,
    151,
    163,
    216
   ],
   [
    28,
    47,
    72,
    130,
    152,
    164,
    217
   ],
   [
    6,
    48,
    49,
    165,
    170,
    171,
    203
   ],
   [
    30,
    49,
    50,
    166,
    172,
    195,
    227
   ],
   [
    31,
    50,
    75,
    167,
    173,
    196,
    228
   ],
   [
    9,
    51,
    52,
    168,
    170,
    173,
    174
   ],
   [
    33,
    52,
    53,
    169,
    175,
    195,
    198
   ],
   [
    34,
    53,
    78,
    170,
    176,
    196,
    199
   ],
   [
    11,
    54,
    55,
    118,
    159,
    171,
    200
   ],
   [
    36,
    55,
    56,
    138,
    160,
    172,
    224
   ],
   [
    37,
    56,
    81,
    139,
    161,
    173,
    225
   ],
   [
    14,
    57,
    58,
    174,
    179,
    180,
    212
   ],
   [
    39,
    58,
    59,
    175,
    181,
    204,
    232
   ],
   [
    40,
    59,
    84,
    122,
    176,
    182,
    205
   ],
   [
    17,
    60,
    61,
    177,
    179,
    182,
    183
   ],
   [
    42,
    61,
    62,
    178,
    184,
    204,
    207
   ],
   [
    43,
    62,
    87,
    179,
    185,
    205,
    208
   ],
   [
    20,
    63,
    64,
    123,
    168,
    180,
    209
   ],
   [
    45,
    64,
    65,
    147,
    169,
    181,
    230
   ],
   [
    46,
    65,
    90,
    148,
    170,
    182,
    231
   ],
   [
    23,
    66,
    67,
    183,
    188,
    189,
    221
   ],
   [
    48,
    67,
    68,
    129,
    184,
    190,
    213
   ],
   [
    49,
    68,
    93,
    130,
    185,
    191,
    214
   ],
   [
    26,
    69,
    70,
    186,
    188,
    191,
    192
   ],
   [
    51,
    70,
    71,
    187,
    193,
    213,
    216
   ],
   [
    52,
    71,
    96,
    188,
    194,
    214,
    217
   ],
   [
    29,
    72,
    73,
    131,
    177,
    189,
    218
   ],
   [
    54,
    73,
    74,
    156,
    178,
    190,
    233
   ],
   [
    55,
    74,
    99,
    127,
    157,
    179,
    191
   ],
   [
    32,
    75,
    76,
    118,
    192,
    197,
    198
   ],
   [
    57,
    76,
    77,
    138,
    193,
    199,
    222
   ],
   [
    58,
    77,
    102,
    139,
    194,
    200,
    223
   ],
   [
    35,
    78,
    79,
    195,
    197,
    200,
    201
   ],
   [
    60,
    79,
    80,
    196,
    202,
    222,
    224
   ],
   [
    61,
    80,
    105,
    197,
    203,
    223,
    225
   ],
   [
    38,
    81,
    82,
    140,
    186,
    198,
    226
   ],
   [
    63,
    82,
    83,
    135,
    165,
    187,
    199
   ],
   [
    64,
    83,
    107,
    136,
    166,
    188,
    200
   ],
   [
    41,
    84,
    85,
    123,
    201,
    206,
    207
   ],
   [
    66,
    85,
    86,
    147,
    202,
    208,
    229
   ],
   [
    67,
    86,
    110,
    119,
    148,
    203,
    209
   ],
   [
    44,
    87,
    88,
    204,
    206,
    209,
    210
   ],
   [
    69,
    88,
    89,
    205,
    211,
    229,
    230
   ],
   [
    70,
    89,
    112,
    119,
    206,
    212,
    231
   ],
   [
    47,
    90,
    91,
    121,
    149,
    195,
    207
   ],
   [
    72,
    91,
    92,
    144,
    174,
    196,
    208
   ],
   [
    73,
    92,
    113,
    145,
    175,
    197,
    209
   ],
   [
    50,
    93,
    94,
    131,
    210,
    215,
    216
   ],
   [
    75,
    94,
    95,
    124,
    156,
    211,
    217
   ],
   [
    76,
    95,
    115,
    125,
    157,
    212,
    218
   ],
   [
    53,
    96,
    97,
    213,
    215,
    218,
    219
   ],
   [
    78,
    97,
    98,
    124,
    214,
    220,
    233
   ],
   [
    7,
    79,
    98,
    125,
    127,
    215,
    221
   ],
   [
    56,
    99,
    100,
    128,
    158,
    204,
    216
   ],
   [
    81,
    100,
    101,
    153,
    183,
    205,
    217
   ],
   [
    82,
    101,
    116,
    154,
    184,
    206,
    218
   ],
   [
    59,
    102,
    103,
    117,
    140,
    219,
    224
   ],
   [
    84,
    103,
    104,
    132,
    165,
    220,
    225
   ],
   [
    12,
    85,
    104,
    133,
    166,
    221,
    226
   ],
   [
    62,
    105,
    106,
    117,
    222,
    226,
    227
   ],
   [
    0,
    87,
    106,
    132,
    135,
    223,
    228
   ],
   [
    65,
    107,
    108,
    137,
    167,
    213,
    224
   ],
   [
    90,
    108,
    109,
    162,
    192,
    214,
    225
   ],
   [
    18,
    91,
    109,
    163,
    193,
    215,
    226
   ],
   [
    68,
    110,
    111,
    120,
    149,
    227,
    230
   ],
   [
    1,
    93,
    111,
    141,
    174,
    228,
    231
   ],
   [
    2,
    71,
    112,
    120,
    121,
    229,
    232
   ],
   [
    74,
    113,
    114,
    146,
    176,
    222,
    230
   ],
   [
    4,
    99,
    114,
    171,
    201,
    223,
    231
   ],
   [
    5,
    77,
    115,
    126,
    158,
    232,
    233
   ],
   [
    10,
    83,
    116,
    155,
    185,
    229,
    233
   ]
  ]
 },
 "distance": {
  "d": 21,
  "X": {
   "value": 21,
   "confidence": "upper_bound",
   "witness": [
    7,
    14,
    18,
    20,
    36,
    80,
    88,
    89,
    91,
    92,
    105,
    110,
    117,
    120,
    121,
    125,
    137,
    142,
    209,
    212,
    224
   ],
   "witness_provenance": {
    "found_by": [
     "@mathysrennela"
    ],
    "date": "2026-10-12",
    "found_at_samples": 2000,
    "survived_samples": 2000,
    "tool": "research/kit/surrogate.lightest_logical",
    "seeds": [
     1018
    ]
   }
  },
  "Z": {
   "value": 21,
   "confidence": "upper_bound",
   "witness": [
    1,
    4,
    5,
    7,
    9,
    24,
    31,
    59,
    60,
    131,
    142,
    146,
    147,
    150,
    160,
    171,
    175,
    179,
    204,
    221,
    228
   ],
   "witness_provenance": {
    "found_by": [
     "@mathysrennela"
    ],
    "date": "2026-10-12",
    "found_at_samples": 2000,
    "survived_samples": 2000,
    "tool": "research/kit/surrogate.lightest_logical",
    "seeds": [
     1019
    ]
   }
  }
 },
 "provenance": {
  "authors": [
   "Lee, Gideon"
  ],
  "construction": "Two-block group-algebra code C(a,b) over C3 x (C13 : C3) = SmallGroup(117, 3) (order 117), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 234; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 16, 19}, b = {0, 15, 88} as element indices. Check weight 7. 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": "2bga-nonabelian"
}