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[[186,10,14]] d ≤
n
186
k
10
d
14
kd²/n
10.538
w
6

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Distance

d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[6, 23, 49, 60, 70, 96, 104, 108, 109, 116, 123, 131, 152, 162]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[0, 6, 12, 18, 33, 41, 50, 61, 70, 96, 104, 110, 117, 143]
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+x2y3, g(x,y)=1+y+x2y-2 on the abelian quotient group Z2/L with twist basis a_1=[0, 31], a_2=[3, 7] (n=2|det[a_1,a_2]|=2*93). 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 93 · Z-checks 93
H_X (93 checks, sparse supports)
[0, 1, 18, 93, 95, 153] [1, 3, 24, 94, 97, 162] [2, 4, 25, 95, 98, 163] [3, 6, 31, 96, 100, 169] [4, 7, 32, 97, 101, 170] [5, 8, 33, 96, 98, 102] [6, 10, 39, 99, 104, 174] [7, 11, 40, 100, 105, 175] [8, 12, 41, 99, 101, 106] [9, 13, 42, 100, 102, 107] [10, 15, 48, 103, 109, 178] [11, 16, 49, 104, 110, 179] [12, 17, 50, 103, 105, 111] [13, 18, 51, 104, 106, 112] [14, 19, 52, 105, 107, 113] [15, 21, 58, 108, 115, 181] [16, 22, 59, 109, 116, 182] [17, 23, 60, 108, 110, 117] [18, 24, 61, 109, 111, 118] [0, 19, 25, 110, 112, 119] [20, 26, 62, 111, 113, 120] [21, 28, 67, 114, 122, 183] [22, 29, 68, 115, 123, 184] [23, 30, 69, 114, 116, 124] [24, 31, 70, 115, 117, 125] [1, 25, 32, 116, 118, 126] [2, 26, 33, 117, 119, 127] [27, 34, 71, 118, 120, 128] [28, 36, 74, 121, 130, 137] [29, 37, 75, 122, 131, 185] [30, 38, 76, 121, 123, 132] [31, 39, 77, 122, 124, 133] [3, 32, 40, 123, 125, 134] [4, 33, 41, 124, 126, 135] [5, 34, 42, 125, 127, 136] [35, 43, 64, 126, 128, 137] [36, 45, 79, 129, 139, 146] [37, 46, 80, 130, 140, 147] [38, 47, 81, 129, 131, 141] [39, 48, 82, 130, 132, 142] [6, 40, 49, 131, 133, 143] [7, 41, 50, 132, 134, 144] [8, 42, 51, 133, 135, 145] [9, 43, 52, 134, 136, 146] [44, 53, 72, 135, 137, 147] [45, 55, 83, 138, 149, 155] [46, 56, 84, 139, 150, 156] [47, 57, 85, 138, 140, 151] [48, 58, 86, 139, 141, 152] [10, 49, 59, 140, 142, 153] [11, 50, 60, 141, 143, 154] [12, 51, 61, 93, 142, 144] [0, 13, 52, 143, 145, 155] [14, 53, 62, 144, 146, 156] [54, 63, 78, 138, 145, 147] [27, 55, 64, 95, 148, 158] [56, 65, 87, 149, 159, 164] [57, 66, 88, 148, 150, 160] [58, 67, 89, 149, 151, 161] [15, 59, 68, 150, 152, 162] [16, 60, 69, 151, 153, 163] [17, 61, 70, 94, 152, 154] [2, 19, 62, 154, 155, 164] [20, 63, 71, 93, 148, 156] [9, 34, 64, 97, 157, 165] [35, 65, 72, 98, 158, 166] [66, 73, 90, 157, 159, 167] [67, 74, 91, 158, 160, 168] [21, 68, 75, 159, 161, 169] [22, 69, 76, 160, 162, 170] [23, 70, 77, 96, 161, 163] [5, 26, 71, 94, 157, 164] [14, 43, 72, 101, 165, 171] [44, 73, 78, 102, 166, 172] [74, 79, 92, 165, 167, 173] [28, 75, 80, 166, 168, 174] [29, 76, 81, 167, 169, 175] [30, 77, 82, 99, 168, 170] [20, 53, 78, 106, 171, 176] [54, 79, 83, 107, 172, 177] [36, 80, 84, 171, 173, 178] [37, 81, 85, 172, 174, 179] [38, 82, 86, 103, 173, 175] [27, 63, 83, 112, 176, 180] [45, 84, 87, 113, 177, 181] [46, 85, 88, 176, 178, 182] [47, 86, 89, 108, 177, 179] [35, 55, 87, 119, 180, 183] [56, 88, 90, 120, 181, 184] [57, 89, 91, 114, 180, 182] [44, 65, 90, 127, 183, 185] [66, 91, 92, 121, 128, 184] [54, 73, 92, 129, 136, 185]
H_Z (93 checks, sparse supports)
[0, 51, 63, 93, 112, 145] [1, 61, 71, 93, 94, 118] [0, 2, 55, 95, 119, 155] [3, 5, 70, 94, 96, 125] [1, 4, 64, 95, 97, 126] [2, 5, 65, 98, 127, 164] [6, 8, 77, 96, 99, 133] [3, 7, 9, 97, 100, 134] [4, 8, 72, 98, 101, 135] [5, 9, 73, 102, 136, 157] [10, 12, 82, 99, 103, 142] [6, 11, 13, 100, 104, 143] [7, 12, 14, 101, 105, 144] [8, 13, 78, 102, 106, 145] [9, 14, 79, 107, 146, 165] [15, 17, 86, 103, 108, 152] [10, 16, 18, 104, 109, 153] [11, 17, 19, 105, 110, 154] [12, 18, 20, 93, 106, 111] [13, 19, 83, 107, 112, 155] [14, 20, 84, 113, 156, 171] [21, 23, 89, 108, 114, 161] [15, 22, 24, 109, 115, 162] [16, 23, 25, 110, 116, 163] [17, 24, 26, 94, 111, 117] [18, 25, 27, 95, 112, 118] [19, 26, 87, 113, 119, 164] [20, 27, 88, 120, 148, 176] [28, 30, 91, 114, 121, 168] [21, 29, 31, 115, 122, 169] [22, 30, 32, 116, 123, 170] [23, 31, 33, 96, 117, 124] [24, 32, 34, 97, 118, 125] [25, 33, 35, 98, 119, 126] [26, 34, 90, 120, 127, 157] [27, 35, 91, 128, 158, 180] [36, 38, 92, 121, 129, 173] [28, 37, 39, 122, 130, 174] [29, 38, 40, 123, 131, 175] [30, 39, 41, 99, 124, 132] [31, 40, 42, 100, 125, 133] [32, 41, 43, 101, 126, 134] [33, 42, 44, 102, 127, 135] [34, 43, 92, 128, 136, 165] [28, 35, 44, 137, 166, 183] [45, 47, 54, 129, 138, 177] [36, 46, 48, 130, 139, 178] [37, 47, 49, 131, 140, 179] [38, 48, 50, 103, 132, 141] [39, 49, 51, 104, 133, 142] [40, 50, 52, 105, 134, 143] [41, 51, 53, 106, 135, 144] [42, 52, 54, 107, 136, 145] [36, 43, 53, 137, 146, 171] [37, 44, 54, 147, 172, 185] [55, 57, 63, 138, 148, 180] [45, 56, 58, 139, 149, 181] [46, 57, 59, 140, 150, 182] [47, 58, 60, 108, 141, 151] [48, 59, 61, 109, 142, 152] [0, 49, 60, 110, 143, 153] [50, 61, 62, 111, 144, 154] [45, 52, 62, 113, 146, 155] [46, 53, 63, 147, 156, 176] [64, 66, 71, 128, 148, 157] [55, 65, 67, 149, 158, 183] [56, 66, 68, 150, 159, 184] [57, 67, 69, 114, 151, 160] [58, 68, 70, 115, 152, 161] [1, 59, 69, 116, 153, 162] [2, 60, 70, 117, 154, 163] [56, 62, 71, 120, 156, 164] [64, 72, 74, 137, 158, 165] [65, 73, 75, 159, 166, 185] [66, 74, 76, 121, 160, 167] [67, 75, 77, 122, 161, 168] [3, 68, 76, 123, 162, 169] [4, 69, 77, 124, 163, 170] [72, 78, 80, 147, 166, 171] [73, 79, 81, 129, 167, 172] [74, 80, 82, 130, 168, 173] [6, 75, 81, 131, 169, 174] [7, 76, 82, 132, 170, 175] [78, 83, 85, 138, 172, 176] [79, 84, 86, 139, 173, 177] [10, 80, 85, 140, 174, 178] [11, 81, 86, 141, 175, 179] [83, 87, 89, 149, 177, 180] [15, 84, 88, 150, 178, 181] [16, 85, 89, 151, 179, 182] [21, 87, 90, 159, 181, 183] [22, 88, 91, 160, 182, 184] [29, 90, 92, 167, 184, 185]
raw submission JSON