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[[98,6,12]] d =
n
98
k
6
d
12
kd²/n
8.816
w
6

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Distance

d_X 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[1, 4, 11, 16, 36, 46, 52, 59, 60, 64, 90, 93]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[0, 21, 23, 34, 40, 60, 66, 79, 82, 85, 89, 95]
certificate exact, d = 12 · CryptoMiniSat 5.14 SAT
X: no logical < 12 exists (CryptoMiniSat, XOR + sequential-counter cardinality); Z: no logical < 12 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+x-1y2, g(x,y)=1+y+x-2y-1 on the abelian quotient group Z2/L with twist basis a_1=[0, 7], a_2=[7, 0] (n=2|det[a_1,a_2]|=2*49). 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 49 · Z-checks 49
H_X (49 checks, sparse supports)
[0, 1, 34, 49, 51, 96] [1, 3, 5, 50, 53, 97] [2, 4, 39, 51, 54, 64] [3, 6, 8, 52, 56, 76] [4, 7, 9, 53, 57, 70] [5, 8, 43, 54, 58, 71] [6, 10, 12, 55, 60, 82] [7, 11, 13, 49, 56, 61] [8, 12, 14, 57, 62, 77] [9, 13, 46, 58, 63, 78] [10, 15, 17, 59, 65, 87] [11, 16, 18, 50, 60, 66] [12, 17, 19, 51, 61, 67] [13, 18, 20, 62, 68, 83] [14, 19, 48, 63, 69, 84] [15, 21, 23, 64, 71, 91] [16, 22, 24, 52, 65, 72] [17, 23, 25, 53, 66, 73] [18, 24, 26, 54, 67, 74] [19, 25, 27, 68, 75, 88] [20, 21, 26, 69, 76, 89] [0, 21, 29, 70, 77, 94] [22, 28, 30, 55, 71, 78] [23, 29, 31, 56, 72, 79] [24, 30, 32, 57, 73, 80] [25, 31, 33, 58, 74, 81] [0, 26, 32, 75, 82, 92] [27, 28, 33, 49, 76, 93] [2, 28, 35, 59, 77, 83] [29, 34, 36, 60, 78, 84] [30, 35, 37, 61, 79, 85] [31, 36, 38, 62, 80, 86] [1, 32, 37, 63, 81, 87] [2, 33, 38, 50, 82, 95] [5, 34, 40, 65, 83, 88] [35, 39, 41, 66, 84, 89] [36, 40, 42, 67, 85, 90] [3, 37, 41, 68, 86, 91] [4, 38, 42, 52, 69, 87] [9, 39, 44, 72, 88, 92] [40, 43, 45, 73, 89, 93] [6, 41, 44, 74, 90, 94] [7, 42, 45, 55, 75, 91] [14, 43, 47, 79, 92, 95] [10, 44, 46, 80, 93, 96] [11, 45, 47, 59, 81, 94] [15, 20, 46, 85, 95, 97] [16, 47, 48, 64, 86, 96] [22, 27, 48, 70, 90, 97]
H_Z (49 checks, sparse supports)
[0, 7, 27, 49, 70, 75] [1, 11, 33, 49, 50, 81] [0, 2, 12, 51, 77, 82] [3, 16, 38, 50, 52, 86] [1, 4, 17, 51, 53, 87] [2, 5, 18, 50, 54, 83] [6, 22, 42, 52, 55, 90] [3, 7, 23, 53, 56, 91] [4, 8, 24, 52, 54, 57] [5, 9, 25, 53, 58, 88] [10, 28, 45, 55, 59, 93] [6, 11, 29, 56, 60, 94] [7, 12, 30, 55, 57, 61] [8, 13, 31, 56, 58, 62] [9, 14, 32, 57, 63, 92] [2, 15, 47, 59, 64, 95] [10, 16, 34, 60, 65, 96] [11, 17, 35, 59, 61, 66] [12, 18, 36, 60, 62, 67] [13, 19, 37, 61, 63, 68] [14, 20, 38, 62, 69, 95] [4, 21, 48, 64, 69, 70] [5, 15, 22, 65, 71, 97] [16, 23, 39, 64, 66, 72] [17, 24, 40, 65, 67, 73] [18, 25, 41, 66, 68, 74] [19, 26, 42, 67, 69, 75] [3, 20, 27, 68, 76, 97] [8, 21, 28, 71, 76, 77] [9, 22, 29, 70, 72, 78] [23, 30, 43, 71, 73, 79] [24, 31, 44, 72, 74, 80] [25, 32, 45, 73, 75, 81] [6, 26, 33, 74, 76, 82] [13, 28, 34, 49, 78, 83] [14, 29, 35, 77, 79, 84] [30, 36, 46, 78, 80, 85] [31, 37, 47, 79, 81, 86] [10, 32, 38, 80, 82, 87] [19, 34, 39, 51, 84, 88] [20, 35, 40, 83, 85, 89] [36, 41, 48, 84, 86, 90] [15, 37, 42, 85, 87, 91] [26, 39, 43, 54, 89, 92] [27, 40, 44, 88, 90, 93] [21, 41, 45, 89, 91, 94] [33, 43, 46, 58, 93, 95] [0, 44, 47, 92, 94, 96] [1, 46, 48, 63, 96, 97]
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