{
 "schema_version": "0.2",
 "name": "[[170,16,13]] two-block code over C85",
 "code_type": "CSS",
 "n": 170,
 "k": 16,
 "checks": {
  "X": [
   [
    0,
    17,
    47,
    81,
    85,
    154,
    169
   ],
   [
    1,
    21,
    51,
    83,
    86,
    158,
    159
   ],
   [
    2,
    3,
    22,
    52,
    87,
    95,
    159
   ],
   [
    3,
    25,
    55,
    84,
    88,
    162,
    163
   ],
   [
    4,
    6,
    26,
    56,
    85,
    89,
    163
   ],
   [
    5,
    7,
    27,
    57,
    85,
    90,
    100
   ],
   [
    6,
    9,
    39,
    74,
    91,
    165,
    166
   ],
   [
    7,
    10,
    30,
    60,
    86,
    92,
    166
   ],
   [
    8,
    11,
    31,
    61,
    86,
    87,
    93
   ],
   [
    9,
    12,
    32,
    62,
    87,
    94,
    105
   ],
   [
    10,
    13,
    43,
    78,
    95,
    167,
    168
   ],
   [
    0,
    11,
    14,
    44,
    88,
    96,
    168
   ],
   [
    12,
    15,
    35,
    65,
    88,
    89,
    97
   ],
   [
    13,
    16,
    36,
    66,
    89,
    90,
    98
   ],
   [
    14,
    17,
    37,
    67,
    90,
    99,
    110
   ],
   [
    1,
    15,
    18,
    48,
    91,
    100,
    169
   ],
   [
    2,
    16,
    19,
    49,
    91,
    92,
    101
   ],
   [
    17,
    20,
    40,
    70,
    92,
    93,
    102
   ],
   [
    18,
    21,
    41,
    71,
    93,
    94,
    103
   ],
   [
    19,
    22,
    42,
    72,
    94,
    104,
    115
   ],
   [
    4,
    20,
    23,
    53,
    95,
    96,
    105
   ],
   [
    5,
    21,
    24,
    54,
    96,
    97,
    106
   ],
   [
    22,
    25,
    45,
    75,
    97,
    98,
    107
   ],
   [
    23,
    26,
    46,
    76,
    98,
    99,
    108
   ],
   [
    24,
    27,
    47,
    77,
    99,
    109,
    120
   ],
   [
    8,
    25,
    28,
    58,
    100,
    101,
    110
   ],
   [
    9,
    26,
    29,
    59,
    101,
    102,
    111
   ],
   [
    27,
    30,
    50,
    79,
    102,
    103,
    112
   ],
   [
    28,
    31,
    51,
    80,
    103,
    104,
    113
   ],
   [
    29,
    32,
    52,
    81,
    104,
    114,
    125
   ],
   [
    13,
    30,
    33,
    63,
    105,
    106,
    115
   ],
   [
    14,
    31,
    34,
    64,
    106,
    107,
    116
   ],
   [
    32,
    35,
    55,
    82,
    107,
    108,
    117
   ],
   [
    33,
    36,
    56,
    83,
    108,
    109,
    118
   ],
   [
    3,
    34,
    37,
    57,
    109,
    119,
    130
   ],
   [
    18,
    35,
    38,
    68,
    110,
    111,
    120
   ],
   [
    19,
    36,
    39,
    69,
    111,
    112,
    121
   ],
   [
    37,
    40,
    60,
    84,
    112,
    113,
    122
   ],
   [
    6,
    38,
    41,
    61,
    113,
    114,
    123
   ],
   [
    7,
    39,
    42,
    62,
    114,
    124,
    135
   ],
   [
    23,
    40,
    43,
    73,
    115,
    116,
    125
   ],
   [
    24,
    41,
    44,
    74,
    116,
    117,
    126
   ],
   [
    10,
    42,
    45,
    65,
    117,
    118,
    127
   ],
   [
    11,
    43,
    46,
    66,
    118,
    119,
    128
   ],
   [
    12,
    44,
    47,
    67,
    119,
    129,
    140
   ],
   [
    28,
    45,
    48,
    78,
    120,
    121,
    130
   ],
   [
    0,
    29,
    46,
    49,
    121,
    122,
    131
   ],
   [
    15,
    47,
    50,
    70,
    122,
    123,
    132
   ],
   [
    16,
    48,
    51,
    71,
    123,
    124,
    133
   ],
   [
    17,
    49,
    52,
    72,
    124,
    134,
    145
   ],
   [
    1,
    33,
    50,
    53,
    125,
    126,
    135
   ],
   [
    2,
    34,
    51,
    54,
    126,
    127,
    136
   ],
   [
    20,
    52,
    55,
    75,
    127,
    128,
    137
   ],
   [
    21,
    53,
    56,
    76,
    128,
    129,
    138
   ],
   [
    22,
    54,
    57,
    77,
    129,
    139,
    150
   ],
   [
    4,
    38,
    55,
    58,
    130,
    131,
    140
   ],
   [
    5,
    39,
    56,
    59,
    131,
    132,
    141
   ],
   [
    25,
    57,
    60,
    79,
    132,
    133,
    142
   ],
   [
    26,
    58,
    61,
    80,
    133,
    134,
    143
   ],
   [
    27,
    59,
    62,
    81,
    134,
    144,
    155
   ],
   [
    8,
    43,
    60,
    63,
    135,
    136,
    145
   ],
   [
    9,
    44,
    61,
    64,
    136,
    137,
    146
   ],
   [
    30,
    62,
    65,
    82,
    137,
    138,
    147
   ],
   [
    31,
    63,
    66,
    83,
    138,
    139,
    148
   ],
   [
    3,
    32,
    64,
    67,
    139,
    149,
    160
   ],
   [
    13,
    48,
    65,
    68,
    140,
    141,
    150
   ],
   [
    14,
    49,
    66,
    69,
    141,
    142,
    151
   ],
   [
    35,
    67,
    70,
    84,
    142,
    143,
    152
   ],
   [
    6,
    36,
    68,
    71,
    143,
    144,
    153
   ],
   [
    7,
    37,
    69,
    72,
    144,
    154,
    164
   ],
   [
    18,
    53,
    70,
    73,
    145,
    146,
    155
   ],
   [
    19,
    54,
    71,
    74,
    146,
    147,
    156
   ],
   [
    10,
    40,
    72,
    75,
    147,
    148,
    157
   ],
   [
    11,
    41,
    73,
    76,
    148,
    149,
    158
   ],
   [
    12,
    42,
    74,
    77,
    149,
    159,
    167
   ],
   [
    23,
    58,
    75,
    78,
    150,
    151,
    160
   ],
   [
    0,
    24,
    59,
    76,
    151,
    152,
    161
   ],
   [
    15,
    45,
    77,
    79,
    152,
    153,
    162
   ],
   [
    16,
    46,
    78,
    80,
    153,
    154,
    163
   ],
   [
    1,
    28,
    63,
    79,
    155,
    156,
    164
   ],
   [
    2,
    29,
    64,
    80,
    156,
    157,
    165
   ],
   [
    20,
    50,
    81,
    82,
    157,
    158,
    166
   ],
   [
    4,
    33,
    68,
    82,
    160,
    161,
    167
   ],
   [
    5,
    34,
    69,
    83,
    161,
    162,
    168
   ],
   [
    8,
    38,
    73,
    84,
    164,
    165,
    169
   ]
  ],
  "Z": [
   [
    0,
    4,
    5,
    85,
    96,
    131,
    161
   ],
   [
    1,
    7,
    8,
    86,
    100,
    135,
    164
   ],
   [
    2,
    8,
    9,
    87,
    101,
    136,
    165
   ],
   [
    3,
    11,
    12,
    87,
    88,
    119,
    149
   ],
   [
    4,
    12,
    13,
    89,
    105,
    140,
    167
   ],
   [
    5,
    13,
    14,
    90,
    106,
    141,
    168
   ],
   [
    6,
    15,
    16,
    89,
    91,
    123,
    153
   ],
   [
    7,
    16,
    17,
    90,
    92,
    124,
    154
   ],
   [
    8,
    17,
    18,
    93,
    110,
    145,
    169
   ],
   [
    9,
    18,
    19,
    91,
    94,
    111,
    146
   ],
   [
    2,
    10,
    20,
    92,
    95,
    127,
    157
   ],
   [
    11,
    20,
    21,
    93,
    96,
    128,
    158
   ],
   [
    12,
    21,
    22,
    94,
    97,
    129,
    159
   ],
   [
    13,
    22,
    23,
    95,
    98,
    115,
    150
   ],
   [
    14,
    23,
    24,
    96,
    99,
    116,
    151
   ],
   [
    5,
    15,
    25,
    97,
    100,
    132,
    162
   ],
   [
    16,
    25,
    26,
    98,
    101,
    133,
    163
   ],
   [
    17,
    26,
    27,
    85,
    99,
    102,
    134
   ],
   [
    18,
    27,
    28,
    100,
    103,
    120,
    155
   ],
   [
    19,
    28,
    29,
    101,
    104,
    121,
    156
   ],
   [
    9,
    20,
    30,
    102,
    105,
    137,
    166
   ],
   [
    21,
    30,
    31,
    86,
    103,
    106,
    138
   ],
   [
    22,
    31,
    32,
    87,
    104,
    107,
    139
   ],
   [
    23,
    32,
    33,
    105,
    108,
    125,
    160
   ],
   [
    24,
    33,
    34,
    106,
    109,
    126,
    161
   ],
   [
    14,
    25,
    35,
    88,
    107,
    110,
    142
   ],
   [
    26,
    35,
    36,
    89,
    108,
    111,
    143
   ],
   [
    27,
    36,
    37,
    90,
    109,
    112,
    144
   ],
   [
    28,
    37,
    38,
    110,
    113,
    130,
    164
   ],
   [
    29,
    38,
    39,
    111,
    114,
    131,
    165
   ],
   [
    19,
    30,
    40,
    92,
    112,
    115,
    147
   ],
   [
    31,
    40,
    41,
    93,
    113,
    116,
    148
   ],
   [
    32,
    41,
    42,
    94,
    114,
    117,
    149
   ],
   [
    33,
    42,
    43,
    115,
    118,
    135,
    167
   ],
   [
    34,
    43,
    44,
    116,
    119,
    136,
    168
   ],
   [
    24,
    35,
    45,
    97,
    117,
    120,
    152
   ],
   [
    36,
    45,
    46,
    98,
    118,
    121,
    153
   ],
   [
    37,
    46,
    47,
    99,
    119,
    122,
    154
   ],
   [
    38,
    47,
    48,
    120,
    123,
    140,
    169
   ],
   [
    39,
    48,
    49,
    91,
    121,
    124,
    141
   ],
   [
    29,
    40,
    50,
    102,
    122,
    125,
    157
   ],
   [
    41,
    50,
    51,
    103,
    123,
    126,
    158
   ],
   [
    42,
    51,
    52,
    104,
    124,
    127,
    159
   ],
   [
    43,
    52,
    53,
    95,
    125,
    128,
    145
   ],
   [
    44,
    53,
    54,
    96,
    126,
    129,
    146
   ],
   [
    34,
    45,
    55,
    107,
    127,
    130,
    162
   ],
   [
    46,
    55,
    56,
    108,
    128,
    131,
    163
   ],
   [
    47,
    56,
    57,
    85,
    109,
    129,
    132
   ],
   [
    48,
    57,
    58,
    100,
    130,
    133,
    150
   ],
   [
    49,
    58,
    59,
    101,
    131,
    134,
    151
   ],
   [
    39,
    50,
    60,
    112,
    132,
    135,
    166
   ],
   [
    51,
    60,
    61,
    86,
    113,
    133,
    136
   ],
   [
    52,
    61,
    62,
    87,
    114,
    134,
    137
   ],
   [
    53,
    62,
    63,
    105,
    135,
    138,
    155
   ],
   [
    54,
    63,
    64,
    106,
    136,
    139,
    156
   ],
   [
    44,
    55,
    65,
    88,
    117,
    137,
    140
   ],
   [
    56,
    65,
    66,
    89,
    118,
    138,
    141
   ],
   [
    57,
    66,
    67,
    90,
    119,
    139,
    142
   ],
   [
    58,
    67,
    68,
    110,
    140,
    143,
    160
   ],
   [
    59,
    68,
    69,
    111,
    141,
    144,
    161
   ],
   [
    49,
    60,
    70,
    92,
    122,
    142,
    145
   ],
   [
    61,
    70,
    71,
    93,
    123,
    143,
    146
   ],
   [
    62,
    71,
    72,
    94,
    124,
    144,
    147
   ],
   [
    63,
    72,
    73,
    115,
    145,
    148,
    164
   ],
   [
    64,
    73,
    74,
    116,
    146,
    149,
    165
   ],
   [
    54,
    65,
    75,
    97,
    127,
    147,
    150
   ],
   [
    66,
    75,
    76,
    98,
    128,
    148,
    151
   ],
   [
    67,
    76,
    77,
    99,
    129,
    149,
    152
   ],
   [
    68,
    77,
    78,
    120,
    150,
    153,
    167
   ],
   [
    0,
    69,
    78,
    121,
    151,
    154,
    168
   ],
   [
    59,
    70,
    79,
    102,
    132,
    152,
    155
   ],
   [
    71,
    79,
    80,
    103,
    133,
    153,
    156
   ],
   [
    72,
    80,
    81,
    104,
    134,
    154,
    157
   ],
   [
    1,
    73,
    81,
    125,
    155,
    158,
    169
   ],
   [
    1,
    2,
    74,
    91,
    126,
    156,
    159
   ],
   [
    64,
    75,
    82,
    107,
    137,
    157,
    160
   ],
   [
    76,
    82,
    83,
    108,
    138,
    158,
    161
   ],
   [
    3,
    77,
    83,
    109,
    139,
    159,
    162
   ],
   [
    3,
    4,
    78,
    95,
    130,
    160,
    163
   ],
   [
    69,
    79,
    84,
    112,
    142,
    162,
    164
   ],
   [
    6,
    80,
    84,
    113,
    143,
    163,
    165
   ],
   [
    6,
    7,
    81,
    85,
    114,
    144,
    166
   ],
   [
    10,
    74,
    82,
    117,
    147,
    166,
    167
   ],
   [
    10,
    11,
    83,
    86,
    118,
    148,
    168
   ],
   [
    0,
    15,
    84,
    88,
    122,
    152,
    169
   ]
  ]
 },
 "distance": {
  "d": 13,
  "X": {
   "value": 13,
   "confidence": "upper_bound",
   "witness": [
    1,
    8,
    9,
    53,
    60,
    61,
    62,
    72,
    74,
    81,
    145,
    146,
    166
   ],
   "witness_provenance": {
    "found_by": [
     "@mathysrennela"
    ],
    "date": "2026-10-12",
    "found_at_samples": 2000,
    "survived_samples": 2000,
    "tool": "research/kit/surrogate.lightest_logical",
    "seeds": [
     1015
    ]
   }
  },
  "Z": {
   "value": 13,
   "confidence": "upper_bound",
   "witness": [
    0,
    37,
    57,
    88,
    89,
    90,
    122,
    132,
    141,
    150,
    151,
    154,
    163
   ],
   "witness_provenance": {
    "found_by": [
     "@mathysrennela"
    ],
    "date": "2026-10-12",
    "found_at_samples": 2000,
    "survived_samples": 2000,
    "tool": "research/kit/surrogate.lightest_logical",
    "seeds": [
     1016
    ]
   }
  }
 },
 "provenance": {
  "authors": [
   "Lee, Gideon"
  ],
  "construction": "Two-block group-algebra code C(a,b) over C85 = SmallGroup(85, 1) (order 85), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 170; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 11, 46, 76}, b = {0, 4, 5} 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": "generalized-bicycle"
}