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[[162,8,14]] d ≤
n
162
k
8
d
14
kd²/n
9.679
w
6

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Distance

d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[9, 14, 24, 29, 69, 75, 89, 97, 140, 146, 150, 151, 154, 157]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[2, 4, 7, 44, 58, 77, 79, 80, 96, 115, 130, 138, 148, 161]
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-1y3, g(x,y)=1+y+x3y-1 on the abelian quotient group Z2/L with twist basis a_1=[0, 9], a_2=[9, -3] (n=2|det[a_1,a_2]|=2*81). 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 81 · Z-checks 81
H_X (81 checks, sparse supports)
[0, 1, 36, 81, 83, 146] [1, 3, 9, 82, 85, 151] [2, 4, 45, 83, 86, 87] [3, 6, 13, 84, 88, 155] [4, 7, 14, 85, 89, 91] [5, 8, 53, 86, 90, 92] [6, 10, 18, 87, 92, 158] [7, 11, 19, 88, 93, 96] [8, 12, 20, 89, 94, 97] [9, 13, 60, 90, 95, 98] [10, 15, 24, 91, 97, 160] [11, 16, 25, 92, 98, 102] [12, 17, 26, 93, 99, 103] [13, 18, 27, 94, 100, 104] [14, 19, 66, 95, 101, 105] [15, 21, 31, 96, 103, 161] [16, 22, 32, 97, 104, 109] [17, 23, 33, 98, 105, 110] [18, 24, 34, 99, 106, 111] [19, 25, 35, 100, 107, 112] [20, 26, 71, 101, 108, 113] [21, 28, 39, 86, 102, 110] [22, 29, 40, 103, 111, 117] [23, 30, 41, 104, 112, 118] [24, 31, 42, 105, 113, 119] [25, 32, 43, 106, 114, 120] [26, 33, 44, 107, 115, 121] [27, 34, 75, 108, 116, 122] [28, 36, 47, 89, 109, 118] [29, 37, 48, 90, 110, 119] [30, 38, 49, 111, 120, 126] [31, 39, 50, 112, 121, 127] [32, 40, 51, 113, 122, 128] [33, 41, 52, 114, 123, 129] [0, 34, 42, 115, 124, 130] [35, 43, 78, 116, 125, 131] [9, 36, 54, 93, 117, 126] [37, 45, 55, 94, 118, 127] [38, 46, 56, 95, 119, 128] [39, 47, 57, 120, 129, 134] [40, 48, 58, 121, 130, 135] [41, 49, 59, 122, 131, 136] [1, 42, 50, 123, 132, 137] [2, 43, 51, 124, 133, 138] [44, 52, 80, 81, 125, 139] [14, 45, 61, 99, 126, 134] [46, 53, 62, 100, 127, 135] [47, 54, 63, 101, 128, 136] [48, 55, 64, 129, 137, 141] [49, 56, 65, 130, 138, 142] [3, 50, 57, 131, 139, 143] [4, 51, 58, 132, 140, 144] [5, 52, 59, 82, 133, 145] [20, 53, 67, 106, 134, 141] [54, 60, 68, 107, 135, 142] [55, 61, 69, 108, 136, 143] [56, 62, 70, 137, 144, 147] [6, 57, 63, 138, 145, 148] [7, 58, 64, 139, 146, 149] [8, 59, 65, 84, 140, 150] [27, 60, 72, 114, 141, 147] [61, 66, 73, 115, 142, 148] [62, 67, 74, 116, 143, 149] [10, 63, 68, 144, 150, 152] [11, 64, 69, 145, 151, 153] [12, 65, 70, 87, 146, 154] [35, 66, 76, 123, 147, 152] [67, 71, 77, 124, 148, 153] [15, 68, 72, 125, 149, 154] [16, 69, 73, 150, 155, 156] [17, 70, 74, 91, 151, 157] [44, 71, 79, 132, 152, 156] [21, 72, 75, 133, 153, 157] [22, 73, 76, 81, 154, 158] [23, 74, 77, 96, 155, 159] [0, 28, 75, 140, 156, 159] [29, 76, 78, 82, 157, 160] [30, 77, 79, 83, 102, 158] [2, 37, 78, 84, 159, 161] [38, 79, 80, 85, 109, 160] [5, 46, 80, 88, 117, 161]
H_Z (81 checks, sparse supports)
[0, 44, 73, 81, 115, 156] [1, 52, 76, 81, 82, 123] [0, 2, 77, 83, 124, 159] [3, 59, 78, 82, 84, 131] [1, 4, 79, 83, 85, 132] [2, 5, 21, 86, 133, 161] [2, 6, 65, 84, 87, 138] [3, 7, 80, 85, 88, 139] [4, 8, 28, 86, 89, 140] [5, 9, 29, 82, 90, 117] [4, 10, 70, 87, 91, 144] [5, 6, 11, 88, 92, 145] [7, 12, 36, 89, 93, 146] [8, 13, 37, 84, 90, 94] [9, 14, 38, 85, 95, 126] [7, 15, 74, 91, 96, 149] [8, 10, 16, 92, 97, 150] [9, 11, 17, 93, 98, 151] [12, 18, 45, 87, 94, 99] [13, 19, 46, 88, 95, 100] [14, 20, 47, 89, 101, 134] [11, 21, 77, 96, 102, 153] [12, 15, 22, 97, 103, 154] [13, 16, 23, 98, 104, 155] [14, 17, 24, 91, 99, 105] [18, 25, 53, 92, 100, 106] [19, 26, 54, 93, 101, 107] [20, 27, 55, 94, 108, 141] [16, 28, 79, 102, 109, 156] [17, 21, 29, 103, 110, 157] [18, 22, 30, 104, 111, 158] [19, 23, 31, 96, 105, 112] [20, 24, 32, 97, 106, 113] [25, 33, 60, 98, 107, 114] [26, 34, 61, 99, 108, 115] [27, 35, 62, 100, 116, 147] [22, 36, 80, 81, 109, 117] [23, 28, 37, 110, 118, 159] [24, 29, 38, 111, 119, 160] [25, 30, 39, 102, 112, 120] [26, 31, 40, 103, 113, 121] [27, 32, 41, 104, 114, 122] [33, 42, 66, 105, 115, 123] [34, 43, 67, 106, 116, 124] [35, 44, 68, 107, 125, 152] [30, 36, 45, 83, 118, 126] [31, 37, 46, 119, 127, 161] [32, 38, 47, 109, 120, 128] [33, 39, 48, 110, 121, 129] [34, 40, 49, 111, 122, 130] [35, 41, 50, 112, 123, 131] [42, 51, 71, 113, 124, 132] [43, 52, 72, 114, 125, 133] [39, 45, 53, 86, 127, 134] [40, 46, 54, 117, 128, 135] [41, 47, 55, 118, 129, 136] [42, 48, 56, 119, 130, 137] [43, 49, 57, 120, 131, 138] [44, 50, 58, 121, 132, 139] [51, 59, 75, 122, 133, 140] [48, 53, 60, 90, 135, 141] [49, 54, 61, 126, 136, 142] [50, 55, 62, 127, 137, 143] [51, 56, 63, 128, 138, 144] [52, 57, 64, 129, 139, 145] [0, 58, 65, 130, 140, 146] [56, 60, 66, 95, 142, 147] [57, 61, 67, 134, 143, 148] [58, 62, 68, 135, 144, 149] [59, 63, 69, 136, 145, 150] [1, 64, 70, 137, 146, 151] [63, 66, 71, 101, 148, 152] [64, 67, 72, 141, 149, 153] [65, 68, 73, 142, 150, 154] [3, 69, 74, 143, 151, 155] [69, 71, 75, 108, 153, 156] [70, 72, 76, 147, 154, 157] [6, 73, 77, 148, 155, 158] [74, 75, 78, 116, 157, 159] [10, 76, 79, 152, 158, 160] [15, 78, 80, 125, 160, 161]
raw submission JSON