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[[40,6,5]] d =
n
40
k
6
d
5
kd²/n
3.75
w
6
X/Z
1
g
0.006
r
5.0
layers
2
swaps
98

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Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · 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 5 · witness weight 5 (claimed exact)
witness operator (support, 5 qubits)
[14, 15, 31, 32, 35]
d_Z 5 · witness weight 5 (claimed exact)
witness operator (support, 5 qubits)
[11, 14, 15, 31, 32]
certificate exact, d = 5 · scipy/HiGHS MILP
X: no logical < 5 exists; Z: no logical < 5 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 4 · H_Z 4 (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 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)
trapping sets H_X (1,2)×20 (2,2)×20 (3,2)×40 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 20 (1,4): 20 (2,2): 20 (2,4): 180 (2,6): 80 (3,2): 40 (3,4): 900 (3,6): 1300 (3,8): 500 (3,10): 20
trapping sets H_Z (1,2)×20 (2,2)×20 (3,2)×40 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 20 (1,4): 20 (2,2): 20 (2,4): 180 (2,6): 80 (3,2): 40 (3,4): 900 (3,6): 1300 (3,8): 500 (3,10): 20
witness diameter X 3.1623 · Z 3.1623 (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)

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
X checkZ checkqubit site (20)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 98 nearest-neighbor SWAPs per round in total, at most 5 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 Lin, Hsiang-Ku and Pryadko, Leonid P.
provenance literature baseline
construction Generalized bicycle (cyclic 2BGA) code over C_20: a(x) = 1 + x4, b(x) = 1 + x + x2 + x7.
model classical construction (no AI model)
reference arXiv:2306.16400, arXiv:2203.17216
date 2023-06-28
notes Parameters from the Lin-Pryadko 2BGA enumeration (arXiv:2306.16400, w=6 abelian/GB tier). Explicit polynomials reconstructed by exhaustive weight-6 GB search over C_20 with the repo research kit; distance proven exact by cutoff MILP (scipy/HiGHS). Seeded so the global frontier reflects the published state of the art (issue #52). Layout certification: a verifier-accepted honest bilayer layout (stacked 4x5 grid, 2 qubits per site, layers=2, measured interaction radius ~5.000) 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 generalized 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 20 (max weight 6) · Z-checks 20 (max weight 6)
H_X (20 checks, sparse supports)
[0, 16, 20, 33, 38, 39] [1, 17, 20, 21, 34, 39] [2, 18, 20, 21, 22, 35] [3, 19, 21, 22, 23, 36] [0, 4, 22, 23, 24, 37] [1, 5, 23, 24, 25, 38] [2, 6, 24, 25, 26, 39] [3, 7, 20, 25, 26, 27] [4, 8, 21, 26, 27, 28] [5, 9, 22, 27, 28, 29] [6, 10, 23, 28, 29, 30] [7, 11, 24, 29, 30, 31] [8, 12, 25, 30, 31, 32] [9, 13, 26, 31, 32, 33] [10, 14, 27, 32, 33, 34] [11, 15, 28, 33, 34, 35] [12, 16, 29, 34, 35, 36] [13, 17, 30, 35, 36, 37] [14, 18, 31, 36, 37, 38] [15, 19, 32, 37, 38, 39]
H_Z (20 checks, sparse supports)
[0, 1, 2, 7, 20, 24] [1, 2, 3, 8, 21, 25] [2, 3, 4, 9, 22, 26] [3, 4, 5, 10, 23, 27] [4, 5, 6, 11, 24, 28] [5, 6, 7, 12, 25, 29] [6, 7, 8, 13, 26, 30] [7, 8, 9, 14, 27, 31] [8, 9, 10, 15, 28, 32] [9, 10, 11, 16, 29, 33] [10, 11, 12, 17, 30, 34] [11, 12, 13, 18, 31, 35] [12, 13, 14, 19, 32, 36] [0, 13, 14, 15, 33, 37] [1, 14, 15, 16, 34, 38] [2, 15, 16, 17, 35, 39] [3, 16, 17, 18, 20, 36] [4, 17, 18, 19, 21, 37] [0, 5, 18, 19, 22, 38] [0, 1, 6, 19, 23, 39]
Code ID 40-6-5 · download JSON · raw on GitHub