{
 "schema_version": "0.2",
 "name": "[[123,3,7]] multi-band dense-packed code",
 "code_type": "CSS",
 "n": 123,
 "k": 3,
 "checks": {
  "X": [
   [
    0,
    7
   ],
   [
    1,
    2,
    8,
    9
   ],
   [
    3,
    4,
    10,
    11
   ],
   [
    5,
    6,
    12,
    13
   ],
   [
    7,
    8,
    14,
    15
   ],
   [
    9,
    10,
    16,
    17
   ],
   [
    11,
    12,
    18,
    19
   ],
   [
    13,
    20
   ],
   [
    14,
    21
   ],
   [
    15,
    16,
    22,
    23
   ],
   [
    17,
    18,
    24,
    25
   ],
   [
    19,
    20,
    26,
    27
   ],
   [
    21,
    22,
    28,
    29
   ],
   [
    23,
    24,
    30,
    31
   ],
   [
    25,
    26,
    32,
    33
   ],
   [
    27,
    34,
    35
   ],
   [
    36,
    37
   ],
   [
    28,
    38
   ],
   [
    29,
    30,
    39,
    40
   ],
   [
    31,
    32,
    41,
    42
   ],
   [
    33,
    34,
    43,
    44
   ],
   [
    35,
    36,
    45,
    46
   ],
   [
    38,
    39,
    48,
    49
   ],
   [
    40,
    41,
    50,
    51
   ],
   [
    42,
    43,
    52,
    53
   ],
   [
    44,
    45,
    54,
    55
   ],
   [
    46,
    47,
    56,
    57
   ],
   [
    51,
    52,
    58,
    59
   ],
   [
    53,
    54,
    60,
    61
   ],
   [
    55,
    56,
    62,
    63
   ],
   [
    59,
    60,
    69,
    70
   ],
   [
    61,
    62,
    71,
    72
   ],
   [
    63,
    64,
    73,
    74
   ],
   [
    65,
    75
   ],
   [
    66,
    67,
    76,
    77
   ],
   [
    68,
    69,
    78,
    79
   ],
   [
    70,
    71,
    80,
    81
   ],
   [
    72,
    73,
    82,
    83
   ],
   [
    75,
    76,
    85,
    86
   ],
   [
    77,
    78,
    87,
    88
   ],
   [
    79,
    80,
    89,
    90
   ],
   [
    81,
    82,
    91,
    92
   ],
   [
    83,
    84,
    93,
    94
   ],
   [
    85,
    95
   ],
   [
    86,
    87,
    96,
    97
   ],
   [
    88,
    89,
    98,
    99
   ],
   [
    90,
    91,
    100,
    101
   ],
   [
    92,
    93
   ],
   [
    95,
    96,
    102,
    103
   ],
   [
    97,
    98,
    104,
    105
   ],
   [
    99,
    100,
    106,
    107
   ],
   [
    101,
    108
   ],
   [
    102,
    109
   ],
   [
    103,
    104,
    110,
    111
   ],
   [
    105,
    106,
    112,
    113
   ],
   [
    107,
    108,
    114,
    115
   ],
   [
    109,
    110,
    116,
    117
   ],
   [
    111,
    112,
    118,
    119
   ],
   [
    113,
    114,
    120,
    121
   ],
   [
    115,
    122
   ]
  ],
  "Z": [
   [
    1,
    2
   ],
   [
    3,
    4
   ],
   [
    5,
    6
   ],
   [
    0,
    1,
    7,
    8
   ],
   [
    2,
    3,
    9,
    10
   ],
   [
    4,
    5,
    11,
    12
   ],
   [
    8,
    9,
    15,
    16
   ],
   [
    10,
    11,
    17,
    18
   ],
   [
    12,
    13,
    19,
    20
   ],
   [
    14,
    15,
    21,
    22
   ],
   [
    16,
    17,
    23,
    24
   ],
   [
    18,
    19,
    25,
    26
   ],
   [
    22,
    23,
    29,
    30
   ],
   [
    24,
    25,
    31,
    32
   ],
   [
    26,
    27,
    33,
    34
   ],
   [
    28,
    29,
    38,
    39
   ],
   [
    30,
    31,
    40,
    41
   ],
   [
    32,
    33,
    42,
    43
   ],
   [
    34,
    35,
    44,
    45
   ],
   [
    36,
    37,
    46,
    47
   ],
   [
    39,
    40,
    49,
    50
   ],
   [
    41,
    42,
    51,
    52
   ],
   [
    43,
    44,
    53,
    54
   ],
   [
    45,
    46,
    55,
    56
   ],
   [
    47,
    57
   ],
   [
    48,
    49
   ],
   [
    50,
    51,
    58
   ],
   [
    52,
    53,
    59,
    60
   ],
   [
    54,
    55,
    61,
    62
   ],
   [
    56,
    57,
    63,
    64
   ],
   [
    66,
    67
   ],
   [
    58,
    59,
    68,
    69
   ],
   [
    60,
    61,
    70,
    71
   ],
   [
    62,
    63,
    72,
    73
   ],
   [
    64,
    74
   ],
   [
    65,
    66,
    75,
    76
   ],
   [
    67,
    68,
    77,
    78
   ],
   [
    69,
    70,
    79,
    80
   ],
   [
    71,
    72,
    81,
    82
   ],
   [
    73,
    74,
    83,
    84
   ],
   [
    76,
    77,
    86,
    87
   ],
   [
    78,
    79,
    88,
    89
   ],
   [
    80,
    81,
    90,
    91
   ],
   [
    82,
    83,
    92,
    93
   ],
   [
    84,
    94
   ],
   [
    85,
    86,
    95,
    96
   ],
   [
    87,
    88,
    97,
    98
   ],
   [
    89,
    90,
    99,
    100
   ],
   [
    96,
    97,
    103,
    104
   ],
   [
    98,
    99,
    105,
    106
   ],
   [
    100,
    101,
    107,
    108
   ],
   [
    102,
    103,
    109,
    110
   ],
   [
    104,
    105,
    111,
    112
   ],
   [
    106,
    107,
    113,
    114
   ],
   [
    110,
    111,
    117,
    118
   ],
   [
    112,
    113,
    119,
    120
   ],
   [
    114,
    115,
    121,
    122
   ],
   [
    116,
    117
   ],
   [
    118,
    119
   ],
   [
    120,
    121
   ]
  ]
 },
 "distance": {
  "d": 7,
  "X": {
   "value": 7,
   "confidence": "upper_bound",
   "witness": [
    21,
    22,
    23,
    24,
    27,
    32,
    33
   ]
  },
  "Z": {
   "value": 7,
   "confidence": "upper_bound",
   "witness": [
    4,
    10,
    17,
    24,
    30,
    39,
    48
   ]
  }
 },
 "provenance": {
  "authors": [
   "@mathysrennela"
  ],
  "construction": "multi-band dense-packed surface code, rows=2 m=2 pitch=6 (raised-pitch generalization of arXiv:2511.06758)",
  "references": [],
  "date": "2026-09-01",
  "notes": "",
  "model": "GLM 5.3 Flash",
  "origin": "submission"
 },
 "family": "local-2d-single",
 "locality": {
  "coordinates": [
   [
    0.5,
    0.5
   ],
   [
    0.5,
    1.5
   ],
   [
    0.5,
    2.5
   ],
   [
    0.5,
    3.5
   ],
   [
    0.5,
    4.5
   ],
   [
    0.5,
    5.5
   ],
   [
    0.5,
    6.5
   ],
   [
    1.5,
    0.5
   ],
   [
    1.5,
    1.5
   ],
   [
    1.5,
    2.5
   ],
   [
    1.5,
    3.5
   ],
   [
    1.5,
    4.5
   ],
   [
    1.5,
    5.5
   ],
   [
    1.5,
    6.5
   ],
   [
    2.5,
    0.5
   ],
   [
    2.5,
    1.5
   ],
   [
    2.5,
    2.5
   ],
   [
    2.5,
    3.5
   ],
   [
    2.5,
    4.5
   ],
   [
    2.5,
    5.5
   ],
   [
    2.5,
    6.5
   ],
   [
    3.5,
    0.5
   ],
   [
    3.5,
    1.5
   ],
   [
    3.5,
    2.5
   ],
   [
    3.5,
    3.5
   ],
   [
    3.5,
    4.5
   ],
   [
    3.5,
    5.5
   ],
   [
    3.5,
    6.5
   ],
   [
    4.5,
    0.5
   ],
   [
    4.5,
    1.5
   ],
   [
    4.5,
    2.5
   ],
   [
    4.5,
    3.5
   ],
   [
    4.5,
    4.5
   ],
   [
    4.5,
    5.5
   ],
   [
    4.5,
    6.5
   ],
   [
    4.5,
    7.5
   ],
   [
    4.5,
    8.5
   ],
   [
    4.5,
    9.5
   ],
   [
    5.5,
    0.5
   ],
   [
    5.5,
    1.5
   ],
   [
    5.5,
    2.5
   ],
   [
    5.5,
    3.5
   ],
   [
    5.5,
    4.5
   ],
   [
    5.5,
    5.5
   ],
   [
    5.5,
    6.5
   ],
   [
    5.5,
    7.5
   ],
   [
    5.5,
    8.5
   ],
   [
    5.5,
    9.5
   ],
   [
    6.5,
    0.5
   ],
   [
    6.5,
    1.5
   ],
   [
    6.5,
    2.5
   ],
   [
    6.5,
    3.5
   ],
   [
    6.5,
    4.5
   ],
   [
    6.5,
    5.5
   ],
   [
    6.5,
    6.5
   ],
   [
    6.5,
    7.5
   ],
   [
    6.5,
    8.5
   ],
   [
    6.5,
    9.5
   ],
   [
    7.5,
    3.5
   ],
   [
    7.5,
    4.5
   ],
   [
    7.5,
    5.5
   ],
   [
    7.5,
    6.5
   ],
   [
    7.5,
    7.5
   ],
   [
    7.5,
    8.5
   ],
   [
    7.5,
    9.5
   ],
   [
    8.5,
    0.5
   ],
   [
    8.5,
    1.5
   ],
   [
    8.5,
    2.5
   ],
   [
    8.5,
    3.5
   ],
   [
    8.5,
    4.5
   ],
   [
    8.5,
    5.5
   ],
   [
    8.5,
    6.5
   ],
   [
    8.5,
    7.5
   ],
   [
    8.5,
    8.5
   ],
   [
    8.5,
    9.5
   ],
   [
    9.5,
    0.5
   ],
   [
    9.5,
    1.5
   ],
   [
    9.5,
    2.5
   ],
   [
    9.5,
    3.5
   ],
   [
    9.5,
    4.5
   ],
   [
    9.5,
    5.5
   ],
   [
    9.5,
    6.5
   ],
   [
    9.5,
    7.5
   ],
   [
    9.5,
    8.5
   ],
   [
    9.5,
    9.5
   ],
   [
    10.5,
    0.5
   ],
   [
    10.5,
    1.5
   ],
   [
    10.5,
    2.5
   ],
   [
    10.5,
    3.5
   ],
   [
    10.5,
    4.5
   ],
   [
    10.5,
    5.5
   ],
   [
    10.5,
    6.5
   ],
   [
    10.5,
    7.5
   ],
   [
    10.5,
    8.5
   ],
   [
    10.5,
    9.5
   ],
   [
    11.5,
    0.5
   ],
   [
    11.5,
    1.5
   ],
   [
    11.5,
    2.5
   ],
   [
    11.5,
    3.5
   ],
   [
    11.5,
    4.5
   ],
   [
    11.5,
    5.5
   ],
   [
    11.5,
    6.5
   ],
   [
    12.5,
    0.5
   ],
   [
    12.5,
    1.5
   ],
   [
    12.5,
    2.5
   ],
   [
    12.5,
    3.5
   ],
   [
    12.5,
    4.5
   ],
   [
    12.5,
    5.5
   ],
   [
    12.5,
    6.5
   ],
   [
    13.5,
    0.5
   ],
   [
    13.5,
    1.5
   ],
   [
    13.5,
    2.5
   ],
   [
    13.5,
    3.5
   ],
   [
    13.5,
    4.5
   ],
   [
    13.5,
    5.5
   ],
   [
    13.5,
    6.5
   ],
   [
    14.5,
    0.5
   ],
   [
    14.5,
    1.5
   ],
   [
    14.5,
    2.5
   ],
   [
    14.5,
    3.5
   ],
   [
    14.5,
    4.5
   ],
   [
    14.5,
    5.5
   ],
   [
    14.5,
    6.5
   ]
  ],
  "interaction_radius": 1.4142135623730951,
  "layers": 1
 },
 "circuit": {
  "d_circ": {
   "X": {
    "value": 7,
    "confidence": "upper_bound",
    "witness": [
     8,
     9,
     23,
     44,
     61,
     80,
     102
    ]
   },
   "Z": {
    "value": 7,
    "confidence": "upper_bound",
    "witness": [
     203,
     204,
     233,
     241,
     337,
     359,
     388
    ]
   }
  },
  "rounds": 7,
  "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/measure layers; CX layers are a proper edge coloring of each Tanner graph (4 X + 4 Z CX layers per round, schedule seed 0; hook-aware orders: every CNOT order suffix needs >= d-1 further data errors to complete a logical, layered by min-conflicts search). Screened by RIS on ker(H_dem) and an observable-constrained seeded BP+OSD search; witnesses validated in GF(2).",
  "contributed_by": {
   "by": [
    "@vprusso"
   ],
   "date": "2026-09-18",
   "method": "generic CSS memory schedule: one ancilla per check, CX layers from a bipartite edge coloring, CNOT orders filtered by a hook-error criterion; screened by RIS on ker(H_dem) and an observable-constrained seeded BP+OSD search"
  }
 }
}
