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[[22,2,6]] d =
n
22
k
2
d
6
kd²/n
3.273
w
8
X/Z
1
g
0.0327
r
3.1623
layers
2
swaps
22

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Distance

X/Z asymmetry 1 · d_X = 6, d_Z = 6 · w_X = 8, w_Z = 8 (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)
[5, 8, 9, 16, 17, 20]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[0, 5, 7, 13, 15, 20]
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 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 8 · H_Z 8
qubit degrees H_X 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)
trapping sets H_X (1,2)×11 (2,2)×11 (3,2)×66 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 11 (1,6): 11 (2,2): 11 (2,4): 55 (2,6): 77 (2,8): 22 (3,2): 66 (3,4): 385 (3,6): 594 (3,8): 154 (3,10): 11
trapping sets H_Z (1,2)×11 (2,2)×11 (3,2)×66 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 11 (1,6): 11 (2,2): 11 (2,4): 55 (2,6): 77 (2,8): 22 (3,2): 66 (3,4): 385 (3,6): 594 (3,8): 154 (3,10): 11
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
layout contributed by @dorakingx · Simulated annealing over site assignments on an integer grid with the repo's own research/local2d/fold_layout.search_layout (max-diameter objective, several bounding boxes and seeds); single layer at radius 4 tried first, then two layers at 7. Radius and per-site occupancy re-measured with the verifier's rule. · 2026-09-26
r = 3.162
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 22 nearest-neighbor SWAPs per round in total, at most 3 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 @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Generalized bicycle code (arXiv:2306.16400, Renyu Wang & Leonid P. Pryadko) from github.com/QEC-pages/GB-codes, gb-codes.zip, codes/GB_22_w8_X.mtx and _Z.mtx.
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-19
notes Literature reproduction of a published generalized bicycle code (arXiv:2306.16400, Renyu Wang & Leonid P. Pryadko), from github.com/QEC-pages/GB-codes gb-codes.zip. Checked against the live board: not equivalent to any existing entry.
family generalized bicycle (a tag, not a ranking)
locality 2D-local bilayer (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[22,2,6]] — generalized bicycle code on Z_{ell} (Wang–Pryadko)

Direction & hypothesis

Target: the unrestricted weight-8 frontier for hackathon issue #1155. This is one of the original generalized bicycle codes of Renyu Wang and Leonid P. Pryadko (arXiv:2306.16400, "Distance bounds for generalized bicycle codes"). It was absent from the board at the time of submission, and the trusted validator reports that it advances the weight-8 unrestricted cell. Efficiency kd²/n = 3.273.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts. From github.com/QEC-pages/GB-codes @ eb3113d, gb-codes.zip, the codes/GB_22_w8_X.mtx and codes/GB_22_w8_Z.mtx pair. These are the actual H_X and H_Z parity-check matrices shipped by the authors; no reconstruction from group parameters was needed. The code is a two-block generalized bicycle code over the circulant ring Z_{ell} with ell = 11.

Evidence trail

The trusted candidate gate returned passed: true — structural verification passed (CSS commutation, connectivity, k = 2), and the random-seed refutation found no lighter logical than the claimed distance. Distance is a witness-backed upper bound d ≤ 6, not an exact certificate. The published distance is consistent with this run.

Dead ends

The same repo ships 72 GB codes; most are dominated by existing board entries or already submitted (the 2BGA-codes parameter mining covered a different parameterization). This (n,k) = (22,2) pair survived the board-dominance screen, the within-pool Pareto filter, and the trusted gate.

Tools

DeepSeek V4 Flash 0731 (provenance.model), Zed coding agent; NumPy, the repository submission builder (research/kit/submit.py) and the trusted validator (verify/validate_candidate.py). Public artifacts fetched by immutable commit URL and treated as data. Bounded local CPU run, no paid compute.

Reproduction

Read the two Matrix Market files GB_22_w8_X.mtx and GB_22_w8_Z.mtx from github.com/QEC-pages/GB-codes @ eb3113d, gb-codes.zip. Each is a 0/1 coordinate matrix: header line nchecks n nnz, then i j 1 entries (1-based). Build dense H_X, H_Z, verify CSS commutation H_X·H_Zᵀ = 0, and pass both to the repository submission builder with confidence upper_bound. Save the returned witnesses immediately and run the default trusted candidate gate. Do not replace the returned distance with the paper's number.

Parity checks

X-checks 11 (max weight 8) · Z-checks 11 (max weight 8)
H_X (11 checks, sparse supports)
[0, 10, 11, 14, 17, 18, 20, 21] [0, 1, 11, 12, 15, 18, 19, 21] [1, 2, 11, 12, 13, 16, 19, 20] [2, 3, 12, 13, 14, 17, 20, 21] [3, 4, 11, 13, 14, 15, 18, 21] [4, 5, 11, 12, 14, 15, 16, 19] [5, 6, 12, 13, 15, 16, 17, 20] [6, 7, 13, 14, 16, 17, 18, 21] [7, 8, 11, 14, 15, 17, 18, 19] [8, 9, 12, 15, 16, 18, 19, 20] [9, 10, 13, 16, 17, 19, 20, 21]
H_Z (11 checks, sparse supports)
[0, 1, 2, 4, 5, 8, 11, 12] [1, 2, 3, 5, 6, 9, 12, 13] [2, 3, 4, 6, 7, 10, 13, 14] [0, 3, 4, 5, 7, 8, 14, 15] [1, 4, 5, 6, 8, 9, 15, 16] [2, 5, 6, 7, 9, 10, 16, 17] [0, 3, 6, 7, 8, 10, 17, 18] [0, 1, 4, 7, 8, 9, 18, 19] [1, 2, 5, 8, 9, 10, 19, 20] [0, 2, 3, 6, 9, 10, 20, 21] [0, 1, 3, 4, 7, 10, 11, 21]
Code ID 22-2-6 · download JSON · raw on GitHub