← back to the board
[[146,18,4]] d =
n
146
k
18
d
4
kd²/n
1.973
w
6

Share this result

Distance

d_X 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[35, 57, 64, 130]
d_Z 4 · witness weight 4 (claimed upper_bound)
witness operator (support, 4 qubits)
[56, 100, 121, 129]
certificate exact, d = 4 · CryptoMiniSat 5.14 SAT
X: no logical < 4 exists (CryptoMiniSat, XOR + sequential-counter cardinality); Z: no logical < 4 exists (CryptoMiniSat, XOR + sequential-counter cardinality)

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