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[[144,12,12]] d =
n
144
k
12
d
12
kd²/n
12.0
w
6

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Distance

d_X 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[54, 56, 60, 64, 67, 71, 126, 127, 130, 131, 133, 137]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[0, 10, 17, 72, 76, 78, 81, 82, 83, 84, 85, 138]
certificate exact, d = 12 · CryptoMiniSat 5.14 SAT
X: no logical < 12 exists (CryptoMiniSat, XOR + sequential-counter cardinality); Z: no logical < 12 exists (CryptoMiniSat, XOR + sequential-counter cardinality)

Construction & provenance

authors Bravyi, Sergey and Cross, Andrew W. and Gambetta, Jay M. and Maslov, Dmitri and Rall, Patrick and Yoder, Theodore J.
provenance literature baseline
construction Bivariate bicycle code, orders (12, 6), A=x3 + y2 + y, B=x2 + x + y3 (periodic boundary conditions).
model classical construction (no AI model)
reference arXiv:2308.07915
date 2023-08-15
notes Distance d=12 established in arXiv:2308.07915 (Table 3); witness here is a decoder-found logical operator of that weight (upper bound).
family bivariate bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 6 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

X-checks 72 · Z-checks 72
H_X (72 checks, sparse supports)
[1, 2, 18, 75, 78, 84] [2, 3, 19, 76, 79, 85] [3, 4, 20, 77, 80, 86] [4, 5, 21, 72, 81, 87] [0, 5, 22, 73, 82, 88] [0, 1, 23, 74, 83, 89] [7, 8, 24, 81, 84, 90] [8, 9, 25, 82, 85, 91] [9, 10, 26, 83, 86, 92] [10, 11, 27, 78, 87, 93] [6, 11, 28, 79, 88, 94] [6, 7, 29, 80, 89, 95] [13, 14, 30, 87, 90, 96] [14, 15, 31, 88, 91, 97] [15, 16, 32, 89, 92, 98] [16, 17, 33, 84, 93, 99] [12, 17, 34, 85, 94, 100] [12, 13, 35, 86, 95, 101] [19, 20, 36, 93, 96, 102] [20, 21, 37, 94, 97, 103] [21, 22, 38, 95, 98, 104] [22, 23, 39, 90, 99, 105] [18, 23, 40, 91, 100, 106] [18, 19, 41, 92, 101, 107] [25, 26, 42, 99, 102, 108] [26, 27, 43, 100, 103, 109] [27, 28, 44, 101, 104, 110] [28, 29, 45, 96, 105, 111] [24, 29, 46, 97, 106, 112] [24, 25, 47, 98, 107, 113] [31, 32, 48, 105, 108, 114] [32, 33, 49, 106, 109, 115] [33, 34, 50, 107, 110, 116] [34, 35, 51, 102, 111, 117] [30, 35, 52, 103, 112, 118] [30, 31, 53, 104, 113, 119] [37, 38, 54, 111, 114, 120] [38, 39, 55, 112, 115, 121] [39, 40, 56, 113, 116, 122] [40, 41, 57, 108, 117, 123] [36, 41, 58, 109, 118, 124] [36, 37, 59, 110, 119, 125] [43, 44, 60, 117, 120, 126] [44, 45, 61, 118, 121, 127] [45, 46, 62, 119, 122, 128] [46, 47, 63, 114, 123, 129] [42, 47, 64, 115, 124, 130] [42, 43, 65, 116, 125, 131] [49, 50, 66, 123, 126, 132] [50, 51, 67, 124, 127, 133] [51, 52, 68, 125, 128, 134] [52, 53, 69, 120, 129, 135] [48, 53, 70, 121, 130, 136] [48, 49, 71, 122, 131, 137] [0, 55, 56, 129, 132, 138] [1, 56, 57, 130, 133, 139] [2, 57, 58, 131, 134, 140] [3, 58, 59, 126, 135, 141] [4, 54, 59, 127, 136, 142] [5, 54, 55, 128, 137, 143] [6, 61, 62, 72, 135, 138] [7, 62, 63, 73, 136, 139] [8, 63, 64, 74, 137, 140] [9, 64, 65, 75, 132, 141] [10, 60, 65, 76, 133, 142] [11, 60, 61, 77, 134, 143] [12, 67, 68, 72, 78, 141] [13, 68, 69, 73, 79, 142] [14, 69, 70, 74, 80, 143] [15, 70, 71, 75, 81, 138] [16, 66, 71, 76, 82, 139] [17, 66, 67, 77, 83, 140]
H_Z (72 checks, sparse supports)
[3, 60, 66, 76, 77, 126] [4, 61, 67, 72, 77, 127] [5, 62, 68, 72, 73, 128] [0, 63, 69, 73, 74, 129] [1, 64, 70, 74, 75, 130] [2, 65, 71, 75, 76, 131] [0, 9, 66, 82, 83, 132] [1, 10, 67, 78, 83, 133] [2, 11, 68, 78, 79, 134] [3, 6, 69, 79, 80, 135] [4, 7, 70, 80, 81, 136] [5, 8, 71, 81, 82, 137] [0, 6, 15, 88, 89, 138] [1, 7, 16, 84, 89, 139] [2, 8, 17, 84, 85, 140] [3, 9, 12, 85, 86, 141] [4, 10, 13, 86, 87, 142] [5, 11, 14, 87, 88, 143] [6, 12, 21, 72, 94, 95] [7, 13, 22, 73, 90, 95] [8, 14, 23, 74, 90, 91] [9, 15, 18, 75, 91, 92] [10, 16, 19, 76, 92, 93] [11, 17, 20, 77, 93, 94] [12, 18, 27, 78, 100, 101] [13, 19, 28, 79, 96, 101] [14, 20, 29, 80, 96, 97] [15, 21, 24, 81, 97, 98] [16, 22, 25, 82, 98, 99] [17, 23, 26, 83, 99, 100] [18, 24, 33, 84, 106, 107] [19, 25, 34, 85, 102, 107] [20, 26, 35, 86, 102, 103] [21, 27, 30, 87, 103, 104] [22, 28, 31, 88, 104, 105] [23, 29, 32, 89, 105, 106] [24, 30, 39, 90, 112, 113] [25, 31, 40, 91, 108, 113] [26, 32, 41, 92, 108, 109] [27, 33, 36, 93, 109, 110] [28, 34, 37, 94, 110, 111] [29, 35, 38, 95, 111, 112] [30, 36, 45, 96, 118, 119] [31, 37, 46, 97, 114, 119] [32, 38, 47, 98, 114, 115] [33, 39, 42, 99, 115, 116] [34, 40, 43, 100, 116, 117] [35, 41, 44, 101, 117, 118] [36, 42, 51, 102, 124, 125] [37, 43, 52, 103, 120, 125] [38, 44, 53, 104, 120, 121] [39, 45, 48, 105, 121, 122] [40, 46, 49, 106, 122, 123] [41, 47, 50, 107, 123, 124] [42, 48, 57, 108, 130, 131] [43, 49, 58, 109, 126, 131] [44, 50, 59, 110, 126, 127] [45, 51, 54, 111, 127, 128] [46, 52, 55, 112, 128, 129] [47, 53, 56, 113, 129, 130] [48, 54, 63, 114, 136, 137] [49, 55, 64, 115, 132, 137] [50, 56, 65, 116, 132, 133] [51, 57, 60, 117, 133, 134] [52, 58, 61, 118, 134, 135] [53, 59, 62, 119, 135, 136] [54, 60, 69, 120, 142, 143] [55, 61, 70, 121, 138, 143] [56, 62, 71, 122, 138, 139] [57, 63, 66, 123, 139, 140] [58, 64, 67, 124, 140, 141] [59, 65, 68, 125, 141, 142]
raw submission JSON