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[[140,6,14]] d ≤
n
140
k
6
d
14
kd²/n
8.4
w
6

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Distance

d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[14, 18, 24, 26, 34, 49, 76, 85, 91, 95, 98, 101, 108, 115]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[39, 40, 48, 53, 54, 66, 85, 92, 100, 106, 108, 110, 111, 124]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors Zijian Liang and Ke Liu and Hao Song and Yu-An Chen
provenance literature baseline
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-2y2 on the abelian quotient group Z2/L with twist basis a_1=[0, 7], a_2=[10, 1] (n=2|det[a_1,a_2]|=2*70). Reconstructed from the published polynomials and twist; CSS form H_X=[f|g], H_Z=[gbar|fbar].
model classical construction (no AI model)
reference 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.
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 70 · Z-checks 70
H_X (70 checks, sparse supports)
[0, 1, 43, 70, 72, 126] [1, 3, 49, 71, 74, 130] [2, 4, 50, 72, 75, 131] [0, 3, 6, 73, 75, 77] [4, 7, 55, 74, 78, 134] [5, 8, 56, 75, 79, 135] [1, 6, 10, 76, 78, 81] [2, 7, 11, 77, 79, 82] [8, 12, 60, 78, 83, 137] [9, 13, 61, 79, 84, 138] [3, 10, 15, 80, 82, 86] [4, 11, 16, 81, 83, 87] [5, 12, 17, 82, 84, 88] [13, 18, 64, 83, 89, 139] [14, 19, 65, 84, 90, 105] [6, 15, 21, 85, 87, 92] [7, 16, 22, 86, 88, 93] [8, 17, 23, 87, 89, 94] [9, 18, 24, 88, 90, 95] [19, 25, 67, 89, 96, 112] [20, 26, 68, 90, 97, 113] [10, 21, 28, 91, 93, 99] [11, 22, 29, 92, 94, 100] [12, 23, 30, 93, 95, 101] [13, 24, 31, 94, 96, 102] [14, 25, 32, 95, 97, 103] [26, 33, 69, 96, 104, 119] [27, 34, 35, 70, 97, 120] [15, 28, 35, 98, 100, 106] [16, 29, 36, 99, 101, 107] [17, 30, 37, 100, 102, 108] [18, 31, 38, 101, 103, 109] [19, 32, 39, 102, 104, 110] [20, 33, 40, 70, 103, 111] [34, 41, 42, 71, 104, 125] [21, 35, 42, 105, 107, 113] [22, 36, 43, 106, 108, 114] [23, 37, 44, 107, 109, 115] [24, 38, 45, 108, 110, 116] [25, 39, 46, 109, 111, 117] [26, 40, 47, 71, 110, 118] [27, 41, 48, 72, 73, 111] [27, 28, 42, 112, 114, 119] [29, 43, 49, 113, 115, 120] [30, 44, 50, 114, 116, 121] [31, 45, 51, 115, 117, 122] [32, 46, 52, 116, 118, 123] [33, 47, 53, 73, 117, 124] [34, 48, 54, 74, 76, 118] [0, 36, 49, 119, 121, 125] [37, 50, 55, 120, 122, 126] [38, 51, 56, 121, 123, 127] [39, 52, 57, 122, 124, 128] [40, 53, 58, 76, 123, 129] [41, 54, 59, 77, 80, 124] [2, 44, 55, 125, 127, 130] [45, 56, 60, 126, 128, 131] [46, 57, 61, 127, 129, 132] [47, 58, 62, 80, 128, 133] [48, 59, 63, 81, 85, 129] [5, 51, 60, 130, 132, 134] [52, 61, 64, 131, 133, 135] [53, 62, 65, 85, 132, 136] [54, 63, 66, 86, 91, 133] [9, 57, 64, 134, 136, 137] [58, 65, 67, 91, 135, 138] [59, 66, 68, 92, 98, 136] [14, 62, 67, 98, 137, 139] [63, 68, 69, 99, 105, 138] [20, 66, 69, 106, 112, 139]
H_Z (70 checks, sparse supports)
[0, 27, 33, 70, 73, 119] [1, 34, 40, 70, 71, 76] [0, 2, 41, 72, 77, 125] [3, 41, 47, 71, 73, 80] [1, 4, 48, 72, 74, 81] [2, 3, 5, 75, 82, 130] [6, 48, 53, 73, 76, 85] [3, 7, 54, 74, 77, 86] [4, 6, 8, 75, 78, 87] [5, 7, 9, 79, 88, 134] [10, 54, 58, 76, 80, 91] [6, 11, 59, 77, 81, 92] [7, 10, 12, 78, 82, 93] [8, 11, 13, 79, 83, 94] [9, 12, 14, 84, 95, 137] [15, 59, 62, 80, 85, 98] [10, 16, 63, 81, 86, 99] [11, 15, 17, 82, 87, 100] [12, 16, 18, 83, 88, 101] [13, 17, 19, 84, 89, 102] [14, 18, 20, 90, 103, 139] [21, 63, 65, 85, 91, 105] [15, 22, 66, 86, 92, 106] [16, 21, 23, 87, 93, 107] [17, 22, 24, 88, 94, 108] [18, 23, 25, 89, 95, 109] [19, 24, 26, 90, 96, 110] [20, 25, 27, 97, 111, 112] [28, 66, 67, 91, 98, 112] [21, 29, 68, 92, 99, 113] [22, 28, 30, 93, 100, 114] [23, 29, 31, 94, 101, 115] [24, 30, 32, 95, 102, 116] [25, 31, 33, 96, 103, 117] [26, 32, 34, 97, 104, 118] [14, 35, 68, 97, 98, 105] [28, 36, 69, 99, 106, 119] [29, 35, 37, 100, 107, 120] [30, 36, 38, 101, 108, 121] [31, 37, 39, 102, 109, 122] [32, 38, 40, 103, 110, 123] [33, 39, 41, 104, 111, 124] [19, 42, 69, 104, 105, 112] [20, 35, 43, 70, 106, 113] [36, 42, 44, 107, 114, 125] [37, 43, 45, 108, 115, 126] [38, 44, 46, 109, 116, 127] [39, 45, 47, 110, 117, 128] [40, 46, 48, 111, 118, 129] [26, 42, 49, 71, 113, 119] [27, 43, 50, 72, 114, 120] [44, 49, 51, 115, 121, 130] [45, 50, 52, 116, 122, 131] [46, 51, 53, 117, 123, 132] [47, 52, 54, 118, 124, 133] [34, 49, 55, 74, 120, 125] [0, 50, 56, 75, 121, 126] [51, 55, 57, 122, 127, 134] [52, 56, 58, 123, 128, 135] [53, 57, 59, 124, 129, 136] [1, 55, 60, 78, 126, 130] [2, 56, 61, 79, 127, 131] [57, 60, 62, 128, 132, 137] [58, 61, 63, 129, 133, 138] [4, 60, 64, 83, 131, 134] [5, 61, 65, 84, 132, 135] [62, 64, 66, 133, 136, 139] [8, 64, 67, 89, 135, 137] [9, 65, 68, 90, 136, 138] [13, 67, 69, 96, 138, 139]
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