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[[154,6,16]] d ≤
n
154
k
6
d
16
kd²/n
9.974
w
6

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Distance

d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[0, 2, 5, 15, 30, 35, 40, 53, 65, 69, 98, 135, 138, 141, 148, 151]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[6, 20, 22, 27, 29, 71, 78, 100, 104, 115, 130, 139, 145, 149, 152, 153]
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-1y2, g(x,y)=1+y+y-4 on the abelian quotient group Z2/L with twist basis a_1=[0, 77], a_2=[1, 16] (n=2|det[a_1,a_2]|=2*77). 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 77 · Z-checks 77
H_X (77 checks, sparse supports)
[0, 1, 47, 77, 79, 144] [1, 3, 5, 78, 81, 149] [2, 4, 55, 79, 82, 92] [3, 6, 8, 80, 84, 152] [4, 7, 9, 81, 85, 98] [5, 8, 62, 82, 86, 99] [6, 10, 12, 83, 88, 153] [7, 11, 13, 84, 89, 105] [8, 12, 14, 85, 90, 106] [9, 13, 68, 86, 91, 107] [10, 15, 17, 87, 93, 128] [11, 16, 18, 88, 94, 113] [12, 17, 19, 89, 95, 114] [13, 18, 20, 90, 96, 115] [14, 19, 73, 77, 91, 97] [15, 21, 23, 92, 99, 135] [16, 22, 24, 93, 100, 121] [17, 23, 25, 94, 101, 122] [18, 24, 26, 95, 102, 123] [19, 25, 27, 78, 96, 103] [20, 26, 70, 79, 97, 104] [21, 28, 30, 98, 106, 141] [22, 29, 31, 99, 107, 129] [23, 30, 32, 100, 108, 130] [24, 31, 33, 101, 109, 131] [25, 32, 34, 80, 102, 110] [26, 33, 35, 81, 103, 111] [27, 34, 74, 82, 104, 112] [28, 36, 38, 105, 114, 146] [0, 29, 37, 106, 115, 136] [30, 38, 39, 77, 107, 137] [0, 31, 40, 108, 116, 138] [32, 39, 41, 83, 109, 117] [33, 40, 42, 84, 110, 118] [34, 41, 43, 85, 111, 119] [35, 42, 76, 86, 112, 120] [36, 44, 46, 113, 122, 150] [1, 37, 45, 114, 123, 142] [2, 38, 46, 78, 115, 143] [2, 39, 48, 87, 116, 124] [40, 47, 49, 88, 117, 125] [41, 48, 50, 89, 118, 126] [42, 49, 51, 90, 119, 127] [36, 43, 50, 91, 120, 128] [44, 52, 54, 104, 121, 130] [3, 45, 53, 122, 131, 147] [4, 46, 54, 80, 123, 148] [5, 47, 56, 93, 124, 132] [48, 55, 57, 94, 125, 133] [49, 56, 58, 95, 126, 134] [44, 50, 57, 96, 127, 135] [45, 51, 58, 97, 121, 128] [52, 59, 61, 111, 129, 137] [6, 53, 60, 112, 130, 138] [7, 54, 61, 83, 131, 151] [9, 55, 63, 100, 132, 139] [56, 62, 64, 101, 133, 140] [52, 57, 63, 102, 134, 141] [53, 58, 64, 103, 129, 135] [59, 65, 67, 118, 136, 143] [10, 60, 66, 119, 137, 144] [11, 61, 67, 87, 120, 138] [14, 62, 69, 108, 139, 145] [59, 63, 68, 109, 140, 146] [60, 64, 69, 110, 136, 141] [65, 70, 72, 125, 142, 148] [15, 66, 71, 126, 143, 149] [16, 67, 72, 92, 127, 144] [20, 65, 68, 116, 145, 150] [66, 69, 73, 117, 142, 146] [35, 70, 75, 132, 147, 151] [21, 71, 74, 133, 148, 152] [22, 72, 75, 98, 134, 149] [27, 71, 73, 124, 147, 150] [28, 43, 74, 139, 151, 153] [29, 75, 76, 105, 140, 152] [37, 51, 76, 113, 145, 153]
H_Z (77 checks, sparse supports)
[0, 14, 30, 77, 106, 108] [1, 19, 38, 77, 78, 114] [0, 2, 20, 79, 115, 116] [3, 25, 46, 78, 80, 122] [1, 4, 26, 79, 81, 123] [2, 5, 27, 78, 82, 124] [6, 32, 54, 80, 83, 130] [3, 7, 33, 81, 84, 131] [4, 8, 34, 80, 82, 85] [5, 9, 35, 81, 86, 132] [10, 39, 61, 83, 87, 137] [6, 11, 40, 84, 88, 138] [7, 12, 41, 83, 85, 89] [8, 13, 42, 84, 86, 90] [9, 14, 43, 85, 91, 139] [2, 15, 67, 87, 92, 143] [10, 16, 47, 88, 93, 144] [11, 17, 48, 87, 89, 94] [12, 18, 49, 88, 90, 95] [13, 19, 50, 89, 91, 96] [14, 20, 51, 90, 97, 145] [4, 21, 72, 92, 98, 148] [5, 15, 22, 93, 99, 149] [16, 23, 55, 92, 94, 100] [17, 24, 56, 93, 95, 101] [18, 25, 57, 94, 96, 102] [19, 26, 58, 95, 97, 103] [20, 27, 44, 96, 104, 150] [7, 28, 75, 98, 105, 151] [8, 21, 29, 99, 106, 152] [9, 22, 30, 98, 100, 107] [23, 31, 62, 99, 101, 108] [24, 32, 63, 100, 102, 109] [25, 33, 64, 101, 103, 110] [26, 34, 52, 102, 104, 111] [27, 35, 53, 103, 112, 147] [11, 36, 76, 105, 113, 120] [12, 28, 37, 106, 114, 153] [13, 29, 38, 105, 107, 115] [31, 39, 68, 107, 109, 116] [32, 40, 69, 108, 110, 117] [33, 41, 59, 109, 111, 118] [34, 42, 60, 110, 112, 119] [35, 43, 61, 111, 120, 151] [16, 44, 51, 113, 121, 127] [17, 36, 45, 114, 122, 128] [18, 37, 46, 113, 115, 123] [39, 47, 73, 77, 117, 124] [40, 48, 65, 116, 118, 125] [41, 49, 66, 117, 119, 126] [42, 50, 67, 118, 120, 127] [10, 43, 51, 119, 128, 153] [22, 52, 58, 121, 129, 134] [23, 44, 53, 122, 130, 135] [24, 45, 54, 121, 123, 131] [47, 55, 70, 79, 125, 132] [48, 56, 71, 124, 126, 133] [49, 57, 72, 125, 127, 134] [15, 50, 58, 126, 128, 135] [29, 59, 64, 129, 136, 140] [30, 52, 60, 130, 137, 141] [31, 53, 61, 129, 131, 138] [55, 62, 74, 82, 133, 139] [56, 63, 75, 132, 134, 140] [21, 57, 64, 133, 135, 141] [37, 65, 69, 136, 142, 145] [38, 59, 66, 137, 143, 146] [0, 60, 67, 136, 138, 144] [62, 68, 76, 86, 140, 145] [28, 63, 69, 139, 141, 146] [45, 70, 73, 97, 142, 147] [46, 65, 71, 143, 148, 150] [1, 66, 72, 142, 144, 149] [36, 68, 73, 91, 146, 150] [54, 70, 74, 104, 148, 151] [3, 71, 75, 147, 149, 152] [6, 74, 76, 112, 152, 153]
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