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[[126,28,8]] d =
n
126
k
28
d
8
kd²/n
14.222
w
10

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Distance

d_X 8 · witness weight 8 (claimed exact)
witness operator (support, 8 qubits)
[28, 63, 76, 78, 84, 90, 111, 125]
d_Z 8 · witness weight 8 (claimed exact)
witness operator (support, 8 qubits)
[3, 9, 11, 24, 25, 39, 60, 122]
certificate exact, d = 8 · scipy/HiGHS MILP
X: min over 28 Z-logical cosets = 8, all proven; Z: min over 28 X-logical cosets = 8, all proven

Construction & provenance

authors Panteleev, Pavel and Kalachev, Gleb
provenance literature baseline
construction Generalized bicycle code, circulant size 63, a(x) = 1 + x + x14 + x16 + x22, b(x) = 1 + x3 + x13 + x20 + x42.
model classical construction (no AI model)
reference arXiv:1904.02703
date 2019-04-04
notes Distance certified exact by MILP (HiGHS); every logical-generator coset proven minimal.
family generalized bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight > 8 (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 63 · Z-checks 63
H_X (63 checks, sparse supports)
[0, 1, 14, 16, 22, 63, 66, 76, 83, 105] [1, 2, 15, 17, 23, 64, 67, 77, 84, 106] [2, 3, 16, 18, 24, 65, 68, 78, 85, 107] [3, 4, 17, 19, 25, 66, 69, 79, 86, 108] [4, 5, 18, 20, 26, 67, 70, 80, 87, 109] [5, 6, 19, 21, 27, 68, 71, 81, 88, 110] [6, 7, 20, 22, 28, 69, 72, 82, 89, 111] [7, 8, 21, 23, 29, 70, 73, 83, 90, 112] [8, 9, 22, 24, 30, 71, 74, 84, 91, 113] [9, 10, 23, 25, 31, 72, 75, 85, 92, 114] [10, 11, 24, 26, 32, 73, 76, 86, 93, 115] [11, 12, 25, 27, 33, 74, 77, 87, 94, 116] [12, 13, 26, 28, 34, 75, 78, 88, 95, 117] [13, 14, 27, 29, 35, 76, 79, 89, 96, 118] [14, 15, 28, 30, 36, 77, 80, 90, 97, 119] [15, 16, 29, 31, 37, 78, 81, 91, 98, 120] [16, 17, 30, 32, 38, 79, 82, 92, 99, 121] [17, 18, 31, 33, 39, 80, 83, 93, 100, 122] [18, 19, 32, 34, 40, 81, 84, 94, 101, 123] [19, 20, 33, 35, 41, 82, 85, 95, 102, 124] [20, 21, 34, 36, 42, 83, 86, 96, 103, 125] [21, 22, 35, 37, 43, 63, 84, 87, 97, 104] [22, 23, 36, 38, 44, 64, 85, 88, 98, 105] [23, 24, 37, 39, 45, 65, 86, 89, 99, 106] [24, 25, 38, 40, 46, 66, 87, 90, 100, 107] [25, 26, 39, 41, 47, 67, 88, 91, 101, 108] [26, 27, 40, 42, 48, 68, 89, 92, 102, 109] [27, 28, 41, 43, 49, 69, 90, 93, 103, 110] [28, 29, 42, 44, 50, 70, 91, 94, 104, 111] [29, 30, 43, 45, 51, 71, 92, 95, 105, 112] [30, 31, 44, 46, 52, 72, 93, 96, 106, 113] [31, 32, 45, 47, 53, 73, 94, 97, 107, 114] [32, 33, 46, 48, 54, 74, 95, 98, 108, 115] [33, 34, 47, 49, 55, 75, 96, 99, 109, 116] [34, 35, 48, 50, 56, 76, 97, 100, 110, 117] [35, 36, 49, 51, 57, 77, 98, 101, 111, 118] [36, 37, 50, 52, 58, 78, 99, 102, 112, 119] [37, 38, 51, 53, 59, 79, 100, 103, 113, 120] [38, 39, 52, 54, 60, 80, 101, 104, 114, 121] [39, 40, 53, 55, 61, 81, 102, 105, 115, 122] [40, 41, 54, 56, 62, 82, 103, 106, 116, 123] [0, 41, 42, 55, 57, 83, 104, 107, 117, 124] [1, 42, 43, 56, 58, 84, 105, 108, 118, 125] [2, 43, 44, 57, 59, 63, 85, 106, 109, 119] [3, 44, 45, 58, 60, 64, 86, 107, 110, 120] [4, 45, 46, 59, 61, 65, 87, 108, 111, 121] [5, 46, 47, 60, 62, 66, 88, 109, 112, 122] [0, 6, 47, 48, 61, 67, 89, 110, 113, 123] [1, 7, 48, 49, 62, 68, 90, 111, 114, 124] [0, 2, 8, 49, 50, 69, 91, 112, 115, 125] [1, 3, 9, 50, 51, 63, 70, 92, 113, 116] [2, 4, 10, 51, 52, 64, 71, 93, 114, 117] [3, 5, 11, 52, 53, 65, 72, 94, 115, 118] [4, 6, 12, 53, 54, 66, 73, 95, 116, 119] [5, 7, 13, 54, 55, 67, 74, 96, 117, 120] [6, 8, 14, 55, 56, 68, 75, 97, 118, 121] [7, 9, 15, 56, 57, 69, 76, 98, 119, 122] [8, 10, 16, 57, 58, 70, 77, 99, 120, 123] [9, 11, 17, 58, 59, 71, 78, 100, 121, 124] [10, 12, 18, 59, 60, 72, 79, 101, 122, 125] [11, 13, 19, 60, 61, 63, 73, 80, 102, 123] [12, 14, 20, 61, 62, 64, 74, 81, 103, 124] [0, 13, 15, 21, 62, 65, 75, 82, 104, 125]
H_Z (63 checks, sparse supports)
[0, 21, 43, 50, 60, 63, 104, 110, 112, 125] [1, 22, 44, 51, 61, 63, 64, 105, 111, 113] [2, 23, 45, 52, 62, 64, 65, 106, 112, 114] [0, 3, 24, 46, 53, 65, 66, 107, 113, 115] [1, 4, 25, 47, 54, 66, 67, 108, 114, 116] [2, 5, 26, 48, 55, 67, 68, 109, 115, 117] [3, 6, 27, 49, 56, 68, 69, 110, 116, 118] [4, 7, 28, 50, 57, 69, 70, 111, 117, 119] [5, 8, 29, 51, 58, 70, 71, 112, 118, 120] [6, 9, 30, 52, 59, 71, 72, 113, 119, 121] [7, 10, 31, 53, 60, 72, 73, 114, 120, 122] [8, 11, 32, 54, 61, 73, 74, 115, 121, 123] [9, 12, 33, 55, 62, 74, 75, 116, 122, 124] [0, 10, 13, 34, 56, 75, 76, 117, 123, 125] [1, 11, 14, 35, 57, 63, 76, 77, 118, 124] [2, 12, 15, 36, 58, 64, 77, 78, 119, 125] [3, 13, 16, 37, 59, 63, 65, 78, 79, 120] [4, 14, 17, 38, 60, 64, 66, 79, 80, 121] [5, 15, 18, 39, 61, 65, 67, 80, 81, 122] [6, 16, 19, 40, 62, 66, 68, 81, 82, 123] [0, 7, 17, 20, 41, 67, 69, 82, 83, 124] [1, 8, 18, 21, 42, 68, 70, 83, 84, 125] [2, 9, 19, 22, 43, 63, 69, 71, 84, 85] [3, 10, 20, 23, 44, 64, 70, 72, 85, 86] [4, 11, 21, 24, 45, 65, 71, 73, 86, 87] [5, 12, 22, 25, 46, 66, 72, 74, 87, 88] [6, 13, 23, 26, 47, 67, 73, 75, 88, 89] [7, 14, 24, 27, 48, 68, 74, 76, 89, 90] [8, 15, 25, 28, 49, 69, 75, 77, 90, 91] [9, 16, 26, 29, 50, 70, 76, 78, 91, 92] [10, 17, 27, 30, 51, 71, 77, 79, 92, 93] [11, 18, 28, 31, 52, 72, 78, 80, 93, 94] [12, 19, 29, 32, 53, 73, 79, 81, 94, 95] [13, 20, 30, 33, 54, 74, 80, 82, 95, 96] [14, 21, 31, 34, 55, 75, 81, 83, 96, 97] [15, 22, 32, 35, 56, 76, 82, 84, 97, 98] [16, 23, 33, 36, 57, 77, 83, 85, 98, 99] [17, 24, 34, 37, 58, 78, 84, 86, 99, 100] [18, 25, 35, 38, 59, 79, 85, 87, 100, 101] [19, 26, 36, 39, 60, 80, 86, 88, 101, 102] [20, 27, 37, 40, 61, 81, 87, 89, 102, 103] [21, 28, 38, 41, 62, 82, 88, 90, 103, 104] [0, 22, 29, 39, 42, 83, 89, 91, 104, 105] [1, 23, 30, 40, 43, 84, 90, 92, 105, 106] [2, 24, 31, 41, 44, 85, 91, 93, 106, 107] [3, 25, 32, 42, 45, 86, 92, 94, 107, 108] [4, 26, 33, 43, 46, 87, 93, 95, 108, 109] [5, 27, 34, 44, 47, 88, 94, 96, 109, 110] [6, 28, 35, 45, 48, 89, 95, 97, 110, 111] [7, 29, 36, 46, 49, 90, 96, 98, 111, 112] [8, 30, 37, 47, 50, 91, 97, 99, 112, 113] [9, 31, 38, 48, 51, 92, 98, 100, 113, 114] [10, 32, 39, 49, 52, 93, 99, 101, 114, 115] [11, 33, 40, 50, 53, 94, 100, 102, 115, 116] [12, 34, 41, 51, 54, 95, 101, 103, 116, 117] [13, 35, 42, 52, 55, 96, 102, 104, 117, 118] [14, 36, 43, 53, 56, 97, 103, 105, 118, 119] [15, 37, 44, 54, 57, 98, 104, 106, 119, 120] [16, 38, 45, 55, 58, 99, 105, 107, 120, 121] [17, 39, 46, 56, 59, 100, 106, 108, 121, 122] [18, 40, 47, 57, 60, 101, 107, 109, 122, 123] [19, 41, 48, 58, 61, 102, 108, 110, 123, 124] [20, 42, 49, 59, 62, 103, 109, 111, 124, 125]
raw submission JSON