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[[62,10,6]] d =
n
62
k
10
d
6
kd²/n
5.806
w
6
X/Z
1
g
0.0344
r
3.6056
layers
2
swaps
148

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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)
[15, 18, 20, 28, 36, 51]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[22, 25, 26, 27, 54, 61]
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)×62 (2,4)×465 (3,3)×186 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 62 (2,4): 465 (3,3): 186 (3,5): 4092 (3,7): 620
trapping sets H_Z (1,3)×62 (2,4)×465 (3,3)×186 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 62 (2,4): 465 (3,3): 186 (3,5): 4092 (3,7): 620
witness diameter X 6.0 · Z 4.5826 (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/62-10-6/ · 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)
[110, 112, 325, 331, 387, 398]
d_circ^Z 6 · fault-set witness of 6 mechanisms (claimed upper_bound)
witness fault set (mechanism indices in the committed .dem, 6)
[710, 762, 1030, 1544, 1549, 1792]
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 = 3.606
X checkZ checkqubit site (32)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 148 nearest-neighbor SWAPs per round in total, at most 4 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+x-1y, g(x,y)=1+y+x-1y-1 on the abelian quotient group Z2/L with twist basis a_1=[0, 31], a_2=[1, 13] (n=2|det[a_1,a_2]|=2*31). 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. 2D-local layout added 2026-07-27: simulated annealing over qubit-to-site assignments on a unit-spaced triangular grid (2 layers, capacity 1 per site per layer); measured interaction radius 3.605551 (bilayer); same code, same check matrices.
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 31 (max weight 6) · Z-checks 31 (max weight 6)
H_X (31 checks, sparse supports)
[0, 1, 30, 31, 33, 53] [1, 2, 3, 32, 35, 57] [2, 4, 20, 33, 36, 58] [3, 4, 6, 34, 38, 60] [4, 5, 7, 31, 35, 39] [5, 8, 25, 36, 40, 61] [6, 7, 10, 37, 42, 45] [7, 8, 11, 32, 38, 43] [8, 9, 12, 33, 39, 44] [9, 13, 29, 40, 45, 51] [10, 11, 15, 41, 47, 50] [11, 12, 16, 34, 42, 48] [12, 13, 17, 35, 43, 49] [13, 14, 18, 36, 44, 50] [1, 14, 19, 34, 45, 56] [15, 16, 20, 46, 52, 55] [16, 17, 21, 37, 47, 53] [17, 18, 22, 38, 48, 54] [18, 19, 23, 39, 49, 55] [3, 19, 24, 37, 40, 50] [5, 20, 21, 51, 56, 59] [21, 22, 25, 41, 52, 57] [22, 23, 26, 42, 53, 58] [23, 24, 27, 43, 54, 59] [6, 24, 28, 41, 44, 55] [9, 25, 26, 46, 56, 60] [26, 27, 29, 31, 47, 57] [0, 27, 28, 48, 58, 61] [10, 28, 30, 46, 49, 59] [0, 14, 29, 32, 52, 60] [2, 15, 30, 51, 54, 61]
H_Z (31 checks, sparse supports)
[0, 4, 26, 31, 58, 60] [1, 7, 29, 31, 32, 45] [0, 2, 8, 32, 33, 61] [3, 11, 14, 32, 34, 50] [1, 4, 12, 33, 34, 35] [2, 5, 13, 35, 36, 51] [6, 16, 19, 34, 37, 55] [3, 7, 17, 35, 37, 38] [4, 8, 18, 36, 38, 39] [5, 9, 19, 39, 40, 56] [10, 21, 24, 37, 41, 59] [6, 11, 22, 38, 41, 42] [7, 12, 23, 39, 42, 43] [8, 13, 24, 40, 43, 44] [6, 9, 14, 44, 45, 60] [15, 25, 28, 41, 46, 61] [10, 16, 26, 42, 46, 47] [11, 17, 27, 43, 47, 48] [12, 18, 28, 44, 48, 49] [10, 13, 19, 45, 49, 50] [9, 20, 30, 33, 46, 51] [15, 21, 29, 47, 51, 52] [0, 16, 22, 48, 52, 53] [17, 23, 30, 49, 53, 54] [15, 18, 24, 50, 54, 55] [14, 20, 25, 36, 52, 56] [1, 21, 26, 53, 56, 57] [2, 22, 27, 54, 57, 58] [20, 23, 28, 55, 58, 59] [3, 25, 29, 40, 57, 60] [5, 27, 30, 31, 59, 61]
Code ID 62-10-6 · download JSON · raw on GitHub