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[[60,8,6]] d =
n
60
k
8
d
6
kd²/n
4.8
w
6
X/Z
1
g
0.0042
r
5.831
layers
2
swaps
217

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Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · 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 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[14, 24, 44, 50, 58, 59]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[2, 8, 13, 23, 35, 43]
certificate exact, d = 6 · scipy/HiGHS MILP
X: no logical < 6 exists; Z: no logical < 6 exists

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)×60 (2,4)×450 (3,3)×240 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 60 (2,4): 450 (3,3): 240 (3,5): 3780 (3,7): 600
trapping sets H_Z (1,3)×60 (2,4)×450 (3,3)×240 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 60 (2,4): 450 (3,3): 240 (3,5): 3780 (3,7): 600
witness diameter X 4.4721 · Z 4.2426 (Euclidean support diameter of the stored distance witnesses in the layout; an upper bound on the exhibited logicals’ spread, not a minimum over all logicals)

Circuit tier

syndrome-extraction memory circuits committed under circuits/60-8-6/ · canonical noise recipe, 6 rounds, stim 1.16.0
d_circ ≤ 6 (min over bases; penalty-only, clamped to ≤ d)
d_circ^X 6 · fault-set witness of 6 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 6)
[249, 258, 478, 696, 771, 1220]
d_circ^Z 6 · fault-set witness of 6 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 6)
[232, 235, 394, 396, 592, 750]
no measured logical error rate yet; d_circ is a floor, and the measured tier records the prefactor it cannot see

Verified 2D layout

as measured by the verifier: every check drawn over the submitted coordinates; the interaction radius is the longest dashed pair
r = 5.831
X checkZ checkqubit site (30)2 qubits stacked (2 layers)dashed: the pair setting the interaction radiushover a check to isolate its qubits; click to pin — repeated clicks cycle through overlapping checks; click empty space to release
routing cost 217 nearest-neighbor SWAPs per round in total, at most 8 for one check (heuristic: MST lower bound on the layout, with one lattice step = the minimum qubit spacing 1; not a rank)

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+y-2, g(x,y)=1+y+x2 on the abelian quotient group Z2/L with twist basis a_1=[0, 10], a_2=[3, 3] (n=2|det[a_1,a_2]|=2*30). 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. Layout certification: a verifier-accepted honest bilayer layout (stacked 5x6 grid, 2 qubits per site, layers=2, measured interaction radius ~5.831) was found by @mathysrennela (Aug 2026). It earns the code the local-2d-bilayer class and makes the site's geometric efficiency g = 4kd^2/(n rho^2 r^4) computable. Code authorship unchanged; layout credit to @mathysrennela.
family bivariate bicycle (a tag, not a ranking)
locality 2D-local bilayer (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 30 (max weight 6) · Z-checks 30 (max weight 6)
H_X (30 checks, sparse supports)
[0, 1, 11, 30, 32, 33] [1, 3, 16, 31, 34, 36] [2, 4, 17, 32, 35, 37] [3, 6, 21, 33, 37, 40] [4, 7, 22, 34, 38, 41] [0, 5, 8, 35, 39, 42] [6, 10, 20, 36, 41, 45] [7, 11, 26, 37, 42, 46] [1, 8, 12, 38, 43, 47] [2, 9, 13, 39, 44, 48] [10, 15, 24, 40, 44, 46] [11, 16, 25, 41, 47, 51] [3, 12, 17, 42, 48, 52] [4, 13, 18, 30, 43, 49] [5, 14, 19, 44, 50, 53] [14, 15, 27, 45, 49, 51] [16, 21, 28, 46, 50, 52] [6, 17, 22, 30, 47, 56] [0, 7, 18, 31, 48, 53] [8, 19, 23, 32, 49, 54] [9, 20, 24, 50, 55, 57] [20, 21, 29, 51, 54, 56] [10, 22, 26, 31, 52, 55] [2, 12, 23, 34, 53, 57] [13, 24, 27, 35, 54, 58] [14, 25, 28, 36, 55, 59] [15, 25, 26, 33, 56, 58] [5, 18, 27, 38, 57, 59] [19, 28, 29, 39, 40, 58] [9, 23, 29, 43, 45, 59]
H_Z (30 checks, sparse supports)
[0, 13, 17, 30, 35, 48] [1, 18, 22, 30, 31, 38] [0, 2, 19, 32, 39, 53] [0, 3, 26, 31, 33, 42] [1, 4, 23, 32, 34, 43] [2, 5, 24, 35, 44, 57] [1, 6, 25, 33, 36, 47] [2, 3, 7, 34, 37, 48] [4, 8, 27, 35, 38, 49] [5, 9, 28, 39, 50, 59] [3, 10, 28, 36, 40, 52] [4, 6, 11, 30, 37, 41] [5, 7, 12, 38, 42, 53] [8, 13, 29, 39, 43, 54] [9, 10, 14, 44, 45, 55] [6, 15, 29, 40, 45, 56] [7, 10, 16, 31, 41, 46] [8, 11, 17, 32, 42, 47] [9, 12, 18, 43, 48, 57] [13, 15, 19, 44, 49, 58] [14, 16, 20, 36, 50, 51] [11, 15, 21, 33, 46, 51] [12, 16, 22, 34, 47, 52] [14, 18, 23, 49, 53, 59] [19, 21, 24, 40, 50, 54] [20, 22, 25, 41, 55, 56] [17, 21, 26, 37, 52, 56] [20, 23, 27, 45, 54, 57] [24, 26, 28, 46, 55, 58] [25, 27, 29, 51, 58, 59]
Code ID 60-8-6 · download JSON · raw on GitHub