[[182,6,18]] · QEC Challenge

[[182,6,18]] d ≤

182

kd²/n

10.681

X/Z

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Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)

d_X 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[2, 18, 22, 53, 65, 76, 77, 82, 83, 99, 103, 114, 131, 149, 163, 172, 178, 180]

d_Z 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[0, 3, 9, 13, 18, 20, 24, 49, 64, 67, 77, 90, 111, 120, 123, 148, 159, 180]

certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking

girth H_X 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)

check weights H_X 6 · H_Z 6

qubit degrees H_X 3 · H_Z 3

trapping sets H_X (1,3)×182 (2,4)×1365 (3,3)×182 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,3): 182 (2,4): 1365 (3,3): 182 (3,5): 13104 (3,7): 1820

trapping sets H_Z (1,3)×182 (2,4)×1365 (3,3)×182 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,3): 182 (2,4): 1365 (3,3): 182 (3,5): 13104 (3,7): 1820

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+x⁴y on the abelian quotient group Z²/L with twist basis a_1=[0, 7], a_2=[13, 1] (n=2|det[a_1,a_2]|=2*91). 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 91 (max weight 6) · Z-checks 91 (max weight 6)

H_X (91 checks, sparse supports)

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

H_Z (91 checks, sparse supports)

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

Code ID 182-6-18 · download JSON · raw on GitHub