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[[192,8,16]] d ≤
n
192
k
8
d
16
kd²/n
10.667
w
6

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Distance

d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[8, 27, 50, 60, 66, 69, 81, 101, 103, 107, 109, 115, 137, 152, 158, 166]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[27, 35, 46, 53, 58, 66, 82, 91, 94, 115, 138, 140, 155, 158, 164, 172]
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, 12], a_2=[8, 2] (n=2|det[a_1,a_2]|=2*96). 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 96 · Z-checks 96
H_X (96 checks, sparse supports)
[0, 1, 74, 96, 98, 169] [1, 3, 9, 97, 100, 176] [2, 4, 81, 98, 101, 102] [3, 6, 13, 99, 103, 182] [4, 7, 14, 100, 104, 106] [5, 8, 87, 101, 105, 107] [6, 10, 18, 102, 107, 187] [7, 11, 19, 103, 108, 111] [8, 12, 20, 104, 109, 112] [9, 13, 92, 105, 110, 113] [10, 15, 24, 106, 112, 190] [11, 16, 25, 107, 113, 117] [12, 17, 26, 108, 114, 118] [13, 18, 27, 109, 115, 119] [14, 19, 95, 110, 116, 120] [15, 21, 31, 111, 118, 150] [16, 22, 32, 112, 119, 124] [17, 23, 33, 113, 120, 125] [18, 24, 34, 114, 121, 126] [19, 25, 35, 115, 122, 127] [20, 26, 90, 116, 123, 128] [21, 28, 39, 117, 125, 159] [22, 29, 40, 118, 126, 132] [23, 30, 41, 119, 127, 133] [24, 31, 42, 120, 128, 134] [25, 32, 43, 121, 129, 135] [26, 33, 44, 122, 130, 136] [27, 34, 94, 123, 131, 137] [28, 36, 48, 124, 133, 167] [29, 37, 49, 125, 134, 141] [30, 38, 50, 126, 135, 142] [31, 39, 51, 127, 136, 143] [32, 40, 52, 128, 137, 144] [33, 41, 53, 129, 138, 145] [34, 42, 54, 130, 139, 146] [28, 35, 43, 131, 140, 147] [36, 45, 57, 132, 142, 174] [37, 46, 58, 133, 143, 151] [38, 47, 59, 134, 144, 152] [39, 48, 60, 96, 135, 145] [40, 49, 61, 136, 146, 153] [41, 50, 62, 137, 147, 154] [42, 51, 63, 138, 148, 155] [36, 43, 52, 139, 149, 156] [37, 44, 53, 140, 150, 157] [2, 45, 55, 141, 152, 180] [46, 56, 66, 96, 142, 160] [0, 47, 67, 143, 153, 161] [48, 57, 68, 97, 144, 154] [49, 58, 69, 98, 145, 155] [50, 59, 70, 146, 156, 162] [51, 60, 71, 147, 157, 163] [45, 52, 61, 148, 158, 164] [46, 53, 62, 149, 159, 165] [47, 54, 63, 132, 150, 166] [4, 55, 64, 151, 161, 185] [5, 56, 65, 97, 152, 168] [2, 57, 75, 99, 153, 162] [58, 66, 76, 100, 154, 163] [59, 67, 77, 101, 155, 164] [60, 68, 78, 156, 165, 170] [55, 61, 69, 157, 166, 171] [56, 62, 70, 158, 167, 172] [0, 63, 71, 141, 159, 173] [7, 64, 72, 160, 169, 189] [8, 65, 73, 99, 161, 175] [5, 66, 82, 103, 162, 170] [67, 74, 83, 104, 163, 171] [68, 75, 84, 105, 164, 172] [64, 69, 76, 165, 173, 177] [65, 70, 77, 166, 174, 178] [1, 71, 78, 151, 167, 179] [11, 72, 79, 168, 176, 191] [12, 73, 80, 102, 169, 181] [9, 74, 88, 108, 170, 177] [75, 81, 89, 109, 171, 178] [72, 76, 82, 110, 172, 179] [73, 77, 83, 173, 180, 183] [3, 78, 84, 160, 174, 184] [16, 79, 85, 131, 175, 182] [17, 80, 86, 106, 176, 186] [14, 81, 93, 114, 177, 183] [79, 82, 87, 115, 178, 184] [80, 83, 88, 116, 179, 185] [6, 84, 89, 168, 180, 188] [22, 85, 90, 139, 181, 187] [23, 86, 91, 111, 140, 182] [20, 85, 87, 121, 183, 188] [86, 88, 92, 122, 184, 189] [10, 89, 93, 123, 175, 185] [29, 44, 90, 148, 186, 190] [30, 91, 94, 117, 149, 187] [27, 91, 92, 129, 188, 191] [15, 93, 95, 130, 181, 189] [38, 54, 94, 124, 158, 190] [21, 35, 95, 138, 186, 191]
H_Z (96 checks, sparse supports)
[0, 39, 46, 96, 143, 159] [1, 48, 56, 96, 97, 167] [0, 2, 49, 98, 141, 153] [3, 57, 65, 97, 99, 174] [1, 4, 58, 98, 100, 151] [2, 5, 59, 101, 152, 162] [2, 6, 73, 99, 102, 180] [3, 7, 66, 100, 103, 160] [4, 8, 67, 101, 104, 161] [5, 9, 68, 97, 105, 170] [4, 10, 80, 102, 106, 185] [5, 6, 11, 103, 107, 168] [7, 12, 74, 104, 108, 169] [8, 13, 75, 99, 105, 109] [9, 14, 76, 100, 110, 177] [7, 15, 86, 106, 111, 189] [8, 10, 16, 107, 112, 175] [9, 11, 17, 108, 113, 176] [12, 18, 81, 102, 109, 114] [13, 19, 82, 103, 110, 115] [14, 20, 83, 104, 116, 183] [11, 21, 91, 111, 117, 191] [12, 15, 22, 112, 118, 181] [13, 16, 23, 113, 119, 182] [14, 17, 24, 106, 114, 120] [18, 25, 87, 107, 115, 121] [19, 26, 88, 108, 116, 122] [20, 27, 89, 109, 123, 188] [16, 28, 94, 117, 124, 131] [17, 21, 29, 118, 125, 186] [18, 22, 30, 119, 126, 187] [19, 23, 31, 111, 120, 127] [20, 24, 32, 112, 121, 128] [25, 33, 92, 113, 122, 129] [26, 34, 93, 114, 123, 130] [27, 35, 79, 115, 131, 191] [22, 36, 54, 124, 132, 139] [23, 28, 37, 125, 133, 140] [24, 29, 38, 126, 134, 190] [25, 30, 39, 117, 127, 135] [26, 31, 40, 118, 128, 136] [27, 32, 41, 119, 129, 137] [33, 42, 95, 120, 130, 138] [34, 43, 85, 121, 131, 139] [35, 44, 86, 122, 140, 186] [29, 45, 63, 132, 141, 148] [30, 36, 46, 133, 142, 149] [31, 37, 47, 134, 143, 150] [32, 38, 48, 124, 135, 144] [33, 39, 49, 125, 136, 145] [34, 40, 50, 126, 137, 146] [35, 41, 51, 127, 138, 147] [42, 52, 90, 128, 139, 148] [43, 53, 91, 129, 140, 149] [15, 44, 54, 130, 150, 190] [37, 55, 71, 141, 151, 157] [38, 45, 56, 142, 152, 158] [40, 47, 57, 132, 144, 153] [41, 48, 58, 133, 145, 154] [42, 49, 59, 134, 146, 155] [43, 50, 60, 135, 147, 156] [44, 51, 61, 136, 148, 157] [52, 62, 94, 137, 149, 158] [21, 53, 63, 138, 150, 159] [46, 64, 78, 151, 160, 165] [47, 55, 65, 152, 161, 166] [50, 57, 66, 142, 154, 162] [51, 58, 67, 143, 155, 163] [52, 59, 68, 144, 156, 164] [53, 60, 69, 145, 157, 165] [54, 61, 70, 146, 158, 166] [28, 62, 71, 147, 159, 167] [56, 72, 84, 160, 168, 172] [0, 64, 73, 161, 169, 173] [60, 66, 74, 96, 163, 170] [61, 67, 75, 153, 164, 171] [62, 68, 76, 154, 165, 172] [63, 69, 77, 155, 166, 173] [36, 70, 78, 156, 167, 174] [65, 79, 89, 168, 175, 178] [1, 72, 80, 169, 176, 179] [69, 74, 81, 98, 171, 177] [70, 75, 82, 162, 172, 178] [71, 76, 83, 163, 173, 179] [45, 77, 84, 164, 174, 180] [73, 85, 93, 175, 181, 183] [3, 79, 86, 176, 182, 184] [77, 81, 87, 101, 178, 183] [78, 82, 88, 170, 179, 184] [55, 83, 89, 171, 180, 185] [80, 90, 95, 116, 181, 186] [6, 85, 91, 182, 187, 188] [84, 87, 92, 105, 184, 188] [64, 88, 93, 177, 185, 189] [10, 90, 94, 123, 187, 190] [72, 92, 95, 110, 189, 191]
raw submission JSON