{
 "schema_version": "0.2",
 "name": "[[84,6,10]] twisted-torus BB code (arXiv:2503.03827), two-layer 2D-local layout",
 "code_type": "CSS",
 "n": 84,
 "k": 6,
 "checks": {
  "X": [
   [
    0,
    1,
    23,
    42,
    44,
    78
   ],
   [
    1,
    3,
    29,
    43,
    46,
    81
   ],
   [
    2,
    4,
    30,
    44,
    47,
    82
   ],
   [
    0,
    3,
    6,
    45,
    47,
    49
   ],
   [
    4,
    7,
    35,
    46,
    50,
    83
   ],
   [
    5,
    8,
    36,
    47,
    51,
    69
   ],
   [
    1,
    6,
    10,
    48,
    50,
    53
   ],
   [
    2,
    7,
    11,
    49,
    51,
    54
   ],
   [
    8,
    12,
    39,
    50,
    55,
    75
   ],
   [
    9,
    13,
    40,
    51,
    56,
    76
   ],
   [
    3,
    10,
    15,
    52,
    54,
    58
   ],
   [
    4,
    11,
    16,
    53,
    55,
    59
   ],
   [
    5,
    12,
    17,
    54,
    56,
    60
   ],
   [
    13,
    18,
    41,
    55,
    61,
    80
   ],
   [
    14,
    19,
    27,
    43,
    56,
    62
   ],
   [
    6,
    15,
    21,
    57,
    59,
    64
   ],
   [
    7,
    16,
    22,
    58,
    60,
    65
   ],
   [
    8,
    17,
    23,
    59,
    61,
    66
   ],
   [
    9,
    18,
    24,
    60,
    62,
    67
   ],
   [
    19,
    25,
    33,
    45,
    61,
    68
   ],
   [
    20,
    26,
    34,
    46,
    48,
    62
   ],
   [
    10,
    21,
    27,
    63,
    65,
    70
   ],
   [
    11,
    22,
    28,
    64,
    66,
    71
   ],
   [
    12,
    23,
    29,
    65,
    67,
    72
   ],
   [
    13,
    24,
    30,
    66,
    68,
    73
   ],
   [
    14,
    25,
    31,
    48,
    67,
    74
   ],
   [
    26,
    32,
    38,
    49,
    52,
    68
   ],
   [
    15,
    27,
    33,
    69,
    71,
    76
   ],
   [
    16,
    28,
    34,
    42,
    70,
    72
   ],
   [
    0,
    17,
    29,
    71,
    73,
    77
   ],
   [
    18,
    30,
    35,
    72,
    74,
    78
   ],
   [
    19,
    31,
    36,
    52,
    73,
    79
   ],
   [
    20,
    32,
    37,
    53,
    57,
    74
   ],
   [
    14,
    21,
    33,
    42,
    75,
    80
   ],
   [
    22,
    34,
    38,
    43,
    76,
    77
   ],
   [
    2,
    24,
    35,
    77,
    79,
    81
   ],
   [
    25,
    36,
    39,
    57,
    78,
    82
   ],
   [
    26,
    37,
    40,
    58,
    63,
    79
   ],
   [
    20,
    28,
    38,
    44,
    45,
    80
   ],
   [
    5,
    31,
    39,
    63,
    81,
    83
   ],
   [
    32,
    40,
    41,
    64,
    69,
    82
   ],
   [
    9,
    37,
    41,
    70,
    75,
    83
   ]
  ],
  "Z": [
   [
    0,
    28,
    33,
    42,
    45,
    71
   ],
   [
    1,
    14,
    34,
    42,
    43,
    48
   ],
   [
    0,
    2,
    38,
    44,
    49,
    77
   ],
   [
    3,
    19,
    38,
    43,
    45,
    52
   ],
   [
    1,
    4,
    20,
    44,
    46,
    53
   ],
   [
    2,
    3,
    5,
    47,
    54,
    81
   ],
   [
    6,
    20,
    25,
    45,
    48,
    57
   ],
   [
    3,
    7,
    26,
    46,
    49,
    58
   ],
   [
    4,
    6,
    8,
    47,
    50,
    59
   ],
   [
    5,
    7,
    9,
    51,
    60,
    83
   ],
   [
    10,
    26,
    31,
    48,
    52,
    63
   ],
   [
    6,
    11,
    32,
    49,
    53,
    64
   ],
   [
    7,
    10,
    12,
    50,
    54,
    65
   ],
   [
    8,
    11,
    13,
    51,
    55,
    66
   ],
   [
    9,
    12,
    14,
    56,
    67,
    75
   ],
   [
    15,
    32,
    36,
    52,
    57,
    69
   ],
   [
    10,
    16,
    37,
    53,
    58,
    70
   ],
   [
    11,
    15,
    17,
    54,
    59,
    71
   ],
   [
    12,
    16,
    18,
    55,
    60,
    72
   ],
   [
    13,
    17,
    19,
    56,
    61,
    73
   ],
   [
    14,
    18,
    20,
    62,
    74,
    80
   ],
   [
    21,
    37,
    39,
    57,
    63,
    75
   ],
   [
    15,
    22,
    40,
    58,
    64,
    76
   ],
   [
    16,
    21,
    23,
    42,
    59,
    65
   ],
   [
    17,
    22,
    24,
    60,
    66,
    77
   ],
   [
    18,
    23,
    25,
    61,
    67,
    78
   ],
   [
    19,
    24,
    26,
    62,
    68,
    79
   ],
   [
    5,
    27,
    40,
    56,
    63,
    69
   ],
   [
    21,
    28,
    41,
    64,
    70,
    80
   ],
   [
    22,
    27,
    29,
    43,
    65,
    71
   ],
   [
    23,
    28,
    30,
    44,
    66,
    72
   ],
   [
    24,
    29,
    31,
    67,
    73,
    81
   ],
   [
    25,
    30,
    32,
    68,
    74,
    82
   ],
   [
    8,
    33,
    41,
    61,
    69,
    75
   ],
   [
    9,
    27,
    34,
    62,
    70,
    76
   ],
   [
    29,
    34,
    35,
    46,
    72,
    77
   ],
   [
    0,
    30,
    36,
    47,
    73,
    78
   ],
   [
    31,
    35,
    37,
    74,
    79,
    83
   ],
   [
    13,
    33,
    38,
    68,
    76,
    80
   ],
   [
    1,
    35,
    39,
    50,
    78,
    81
   ],
   [
    2,
    36,
    40,
    51,
    79,
    82
   ],
   [
    4,
    39,
    41,
    55,
    82,
    83
   ]
  ]
 },
 "distance": {
  "d": 10,
  "X": {
   "value": 10,
   "confidence": "upper_bound",
   "witness": [
    0,
    13,
    33,
    35,
    55,
    60,
    61,
    77,
    78,
    83
   ]
  },
  "Z": {
   "value": 10,
   "confidence": "upper_bound",
   "witness": [
    10,
    14,
    15,
    23,
    27,
    42,
    53,
    59,
    72,
    74
   ]
  }
 },
 "provenance": {
  "authors": [
   "Zijian Liang",
   "Ke Liu",
   "Hao Song",
   "Yu-An Chen"
  ],
  "construction": "Twisted-torus bivariate-bicycle code from Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025). Stabilizers f(x,y)=1+x+x^-2, g(x,y)=1+y+x^-2y^2 on the abelian quotient group Z^2/L with twist basis a_1=[0, 14], a_2=[3, -6] (n=2|det[a_1,a_2]|=2*42). Reconstructed from the published polynomials and twist; CSS form H_X=[f|g], H_Z=[gbar|fbar].",
  "references": [
   "arXiv:2503.03827"
  ],
  "date": "2025",
  "notes": "Reconstructed baseline from Liang, Liu, Song, Chen, Generalized toric codes on twisted tori for quantum error correction (PRX Quantum 6, 020357, 2025), arXiv:2503.03827. The [[n,k,d]] parameter set and stabilizer polynomials are published in that paper; this entry reproduces them. Distance is an upper bound per the paper (probabilistic for d>20); the witness is the surrogate logical of that weight. Seeded to fill a board coverage gap at this block size, not a discovery. LAYOUT added 2026-08-20: a two-layer planar layout with measured interaction radius exactly 5.0 (integer-grid sites, min site spacing 1.0, at most 2 qubits per site), placing the code in the local-2d-bilayer class with kd^2/n = 7.14 (previous weight-6 bilayer cell best 6.00, codes/72-12-6.json) and geometric efficiency g = 0.0114. Layout credit and method are recorded in locality.contributed_by; the CODE, its distance claims, and its authorship are unchanged from the literature baseline. A structured product-fold layout derived from the code's Z_42 translation symmetry bottoms out at sqrt(52) ~ 7.21, so the annealed non-product layout is strictly better and likely at or near this code's floor.",
  "origin": "baseline",
  "novelty": "known_parameters"
 },
 "family": "bivariate-bicycle",
 "locality": {
  "coordinates": [
   [
    5.0,
    7.0
   ],
   [
    8.0,
    7.0
   ],
   [
    7.0,
    4.0
   ],
   [
    6.0,
    3.0
   ],
   [
    10.0,
    7.0
   ],
   [
    9.0,
    4.0
   ],
   [
    9.0,
    5.0
   ],
   [
    9.0,
    4.0
   ],
   [
    9.0,
    8.0
   ],
   [
    7.0,
    8.0
   ],
   [
    7.0,
    3.0
   ],
   [
    9.0,
    6.0
   ],
   [
    8.0,
    7.0
   ],
   [
    6.0,
    8.0
   ],
   [
    6.0,
    9.0
   ],
   [
    8.0,
    3.0
   ],
   [
    8.0,
    6.0
   ],
   [
    5.0,
    6.0
   ],
   [
    7.0,
    10.0
   ],
   [
    4.0,
    7.0
   ],
   [
    8.0,
    8.0
   ],
   [
    9.0,
    7.0
   ],
   [
    5.0,
    5.0
   ],
   [
    6.0,
    10.0
   ],
   [
    5.0,
    8.0
   ],
   [
    8.0,
    9.0
   ],
   [
    6.0,
    4.0
   ],
   [
    5.0,
    5.0
   ],
   [
    7.0,
    9.0
   ],
   [
    4.0,
    6.0
   ],
   [
    6.0,
    8.0
   ],
   [
    6.0,
    7.0
   ],
   [
    8.0,
    4.0
   ],
   [
    8.0,
    6.0
   ],
   [
    4.0,
    8.0
   ],
   [
    7.0,
    8.0
   ],
   [
    7.0,
    5.0
   ],
   [
    10.0,
    5.0
   ],
   [
    4.0,
    5.0
   ],
   [
    10.0,
    8.0
   ],
   [
    7.0,
    4.0
   ],
   [
    9.0,
    8.0
   ],
   [
    6.0,
    10.0
   ],
   [
    4.0,
    6.0
   ],
   [
    6.0,
    7.0
   ],
   [
    5.0,
    7.0
   ],
   [
    6.0,
    5.0
   ],
   [
    8.0,
    4.0
   ],
   [
    6.0,
    6.0
   ],
   [
    6.0,
    3.0
   ],
   [
    10.0,
    6.0
   ],
   [
    9.0,
    6.0
   ],
   [
    5.0,
    4.0
   ],
   [
    10.0,
    5.0
   ],
   [
    7.0,
    3.0
   ],
   [
    9.0,
    9.0
   ],
   [
    7.0,
    7.0
   ],
   [
    7.0,
    5.0
   ],
   [
    8.0,
    3.0
   ],
   [
    9.0,
    7.0
   ],
   [
    7.0,
    7.0
   ],
   [
    7.0,
    9.0
   ],
   [
    4.0,
    8.0
   ],
   [
    9.0,
    5.0
   ],
   [
    8.0,
    5.0
   ],
   [
    6.0,
    6.0
   ],
   [
    6.0,
    9.0
   ],
   [
    5.0,
    9.0
   ],
   [
    6.0,
    5.0
   ],
   [
    8.0,
    5.0
   ],
   [
    7.0,
    6.0
   ],
   [
    5.0,
    6.0
   ],
   [
    5.0,
    9.0
   ],
   [
    4.0,
    7.0
   ],
   [
    8.0,
    8.0
   ],
   [
    9.0,
    9.0
   ],
   [
    5.0,
    4.0
   ],
   [
    4.0,
    5.0
   ],
   [
    8.0,
    9.0
   ],
   [
    6.0,
    4.0
   ],
   [
    5.0,
    8.0
   ],
   [
    7.0,
    6.0
   ],
   [
    10.0,
    6.0
   ],
   [
    10.0,
    7.0
   ]
  ],
  "layers": 2,
  "interaction_radius": 5.0,
  "contributed_by": {
   "by": [
    "@willzeng"
   ],
   "date": "2026-08-20",
   "method": "Simulated annealing over integer-grid site assignments (spectral initialization, Metropolis moves with softmax cost, max-focused hill-climb polish); 22 independent runs converged to radius 5.0 or 5.099. Search driven by Claude Fable 5."
  }
 }
}