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[[24,6,4]] d =
n
24
k
6
d
4
kd²/n
4.0
w
6
X/Z
1
g
0.0816
r
2.6458
layers
2
swaps
16

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Distance

X/Z asymmetry 1 · d_X = 4, d_Z = 4 · 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 4 · witness weight 4 (claimed exact)
witness operator (support, 4 qubits)
[1, 7, 13, 19]
d_Z 4 · witness weight 4 (claimed exact)
witness operator (support, 4 qubits)
[1, 7, 13, 19]
certificate exact, d = 4 · scipy/HiGHS MILP
X: no logical < 4 exists; Z: no logical < 4 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)×12 (2,2)×24 (3,2)×108 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 12 (1,4): 12 (2,2): 24 (2,4): 78 (2,6): 60 (3,2): 108 (3,4): 360 (3,6): 616 (3,8): 72 (3,10): 24
trapping sets H_Z (1,2)×12 (2,2)×24 (3,2)×108 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 12 (1,4): 12 (2,2): 24 (2,4): 78 (2,6): 60 (3,2): 108 (3,4): 360 (3,6): 616 (3,8): 72 (3,10): 24
witness diameter X 2.0 · Z 2.0 (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 = 2.646
X checkZ checkqubit site (12)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 16 nearest-neighbor SWAPs per round in total, at most 2 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_12: a(x) = 1 + x3, b(x) = 1 + x + x3 + 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_12 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). 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 2.645751 (bilayer); same code, same check matrices.
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 12 (max weight 6) · Z-checks 12 (max weight 6)
H_X (12 checks, sparse supports)
[0, 9, 12, 17, 21, 23] [1, 10, 12, 13, 18, 22] [2, 11, 13, 14, 19, 23] [0, 3, 12, 14, 15, 20] [1, 4, 13, 15, 16, 21] [2, 5, 14, 16, 17, 22] [3, 6, 15, 17, 18, 23] [4, 7, 12, 16, 18, 19] [5, 8, 13, 17, 19, 20] [6, 9, 14, 18, 20, 21] [7, 10, 15, 19, 21, 22] [8, 11, 16, 20, 22, 23]
H_Z (12 checks, sparse supports)
[0, 1, 3, 7, 12, 15] [1, 2, 4, 8, 13, 16] [2, 3, 5, 9, 14, 17] [3, 4, 6, 10, 15, 18] [4, 5, 7, 11, 16, 19] [0, 5, 6, 8, 17, 20] [1, 6, 7, 9, 18, 21] [2, 7, 8, 10, 19, 22] [3, 8, 9, 11, 20, 23] [0, 4, 9, 10, 12, 21] [1, 5, 10, 11, 13, 22] [0, 2, 6, 11, 14, 23]
Code ID 24-6-4 · download JSON · raw on GitHub