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[[96,4,12]] d =
n
96
k
4
d
12
kd²/n
6.0
w
6

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Distance

d_X 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[18, 21, 25, 28, 31, 43, 47, 63, 68, 85, 87, 90]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[19, 32, 41, 55, 64, 67, 75, 80, 82, 86, 89, 91]
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-2y, g(x,y)=1+y+xy-2 on the abelian quotient group Z2/L with twist basis a_1=[0, 12], a_2=[4, 2] (n=2|det[a_1,a_2]|=2*48). 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 48 · Z-checks 48
H_X (48 checks, sparse supports)
[0, 1, 18, 48, 50, 63] [1, 3, 23, 49, 52, 69] [2, 4, 24, 50, 53, 70] [2, 3, 6, 51, 55, 75] [4, 7, 29, 52, 56, 76] [5, 8, 30, 49, 53, 57] [4, 6, 10, 54, 59, 81] [5, 7, 11, 55, 60, 82] [8, 12, 35, 51, 56, 61] [9, 13, 36, 52, 57, 62] [7, 10, 15, 58, 64, 86] [8, 11, 16, 59, 65, 87] [9, 12, 17, 54, 60, 66] [13, 18, 40, 55, 61, 67] [14, 19, 41, 56, 62, 68] [11, 15, 21, 63, 70, 90] [12, 16, 22, 48, 64, 91] [0, 13, 17, 58, 65, 71] [14, 18, 23, 59, 66, 72] [19, 24, 44, 60, 67, 73] [20, 25, 42, 61, 68, 74] [16, 21, 27, 69, 76, 93] [17, 22, 28, 49, 70, 94] [2, 19, 23, 64, 71, 77] [20, 24, 29, 65, 72, 78] [25, 30, 45, 66, 73, 79] [26, 31, 46, 67, 74, 80] [22, 27, 33, 74, 75, 82] [0, 28, 34, 51, 76, 95] [5, 25, 29, 48, 77, 83] [26, 30, 35, 71, 78, 84] [31, 36, 47, 72, 79, 85] [21, 32, 37, 73, 80, 81] [28, 33, 38, 79, 81, 87] [1, 34, 39, 54, 80, 82] [9, 31, 35, 50, 83, 88] [32, 36, 40, 77, 84, 89] [27, 37, 41, 78, 85, 86] [34, 38, 42, 84, 86, 91] [3, 39, 43, 58, 85, 87] [14, 37, 40, 53, 88, 92] [33, 41, 44, 83, 89, 90] [39, 42, 45, 88, 90, 94] [6, 43, 46, 63, 89, 91] [20, 38, 44, 57, 92, 93] [26, 43, 45, 62, 93, 95] [10, 46, 47, 69, 92, 94] [15, 32, 47, 68, 75, 95]
H_Z (48 checks, sparse supports)
[0, 16, 29, 48, 65, 76] [1, 5, 22, 48, 49, 82] [0, 2, 35, 50, 51, 71] [3, 8, 28, 49, 51, 87] [1, 4, 9, 50, 52, 54] [2, 5, 40, 53, 55, 77] [6, 12, 34, 51, 54, 91] [3, 7, 13, 52, 55, 58] [4, 8, 14, 53, 56, 59] [5, 9, 44, 57, 60, 83] [10, 17, 39, 54, 58, 94] [6, 11, 18, 55, 59, 63] [7, 12, 19, 56, 60, 64] [8, 13, 20, 57, 61, 65] [9, 14, 45, 62, 66, 88] [0, 15, 43, 58, 63, 95] [10, 16, 23, 59, 64, 69] [11, 17, 24, 60, 65, 70] [12, 18, 25, 48, 61, 66] [13, 19, 26, 62, 67, 71] [14, 20, 47, 68, 72, 92] [1, 21, 46, 63, 69, 80] [2, 15, 22, 64, 70, 75] [17, 23, 30, 49, 66, 71] [18, 24, 31, 50, 67, 72] [19, 25, 32, 68, 73, 77] [20, 26, 27, 74, 78, 93] [3, 27, 47, 69, 75, 85] [4, 21, 28, 70, 76, 81] [23, 29, 36, 52, 72, 77] [24, 30, 37, 53, 73, 78] [25, 31, 33, 74, 79, 83] [26, 32, 34, 80, 84, 95] [6, 32, 33, 75, 81, 89] [7, 27, 34, 76, 82, 86] [29, 35, 41, 56, 78, 83] [30, 36, 38, 57, 79, 84] [31, 37, 39, 80, 85, 88] [10, 37, 38, 81, 86, 92] [11, 33, 39, 82, 87, 90] [35, 40, 42, 61, 84, 88] [36, 41, 43, 62, 85, 89] [15, 41, 42, 68, 86, 90] [16, 38, 43, 87, 91, 93] [40, 44, 46, 67, 89, 92] [21, 44, 45, 73, 90, 93] [22, 42, 46, 74, 91, 94] [28, 45, 47, 79, 94, 95]
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