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[[22,2,5]] d =
n
22
k
2
d
5
kd²/n
2.273
w
6
X/Z
1
g
0.0227
r
3.1623
layers
2
swaps
22

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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 upper_bound)
witness operator (support, 5 qubits)
[5, 12, 13, 14, 16]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[3, 5, 6, 7, 14]
certificate exact, d = 5 · scipy/HiGHS cutoff IP
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)×11 (2,2)×33 (3,2)×99 (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 (1,4): 11 (2,2): 33 (2,4): 55 (2,6): 44 (3,2): 99 (3,4): 385 (3,6): 341 (3,8): 44 (3,10): 11
trapping sets H_Z (1,2)×11 (2,2)×33 (3,2)×99 (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 (1,4): 11 (2,2): 33 (2,4): 55 (2,6): 44 (3,2): 99 (3,4): 385 (3,6): 341 (3,8): 44 (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 (11)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 2BGA (arXiv:2306.16400) on SmallGroup(11,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt6wtL2_order11_k2.txt, line 1. nonidentity supports a=[2], b=[2, 3, 6].
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-19
notes Literature reproduction of a published 2BGA code (arXiv:2306.16400; github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c). 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 ≤ 6 (computed)

How this code was found

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

[[22,2,5]] — two-block group-algebra code on SmallGroup(11,1)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 2.27.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts and parameter tables. From github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt6wtL2_order11_k2.txt, line 1. This row is SmallGroup(11,1) with nonidentity GAP supports a=[2], b=[2, 3, 6]. Reconstructed via GAP (Cayley table) and the repository's group-algebra constructor.

Evidence trail

The trusted candidate gate returned passed: true, structural verification passed (CSS, connectivity, k=2), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 5, not an exact certificate. The published distance (d=5) is consistent with this run.

Dead ends

None for this candidate. The overall archive screen found 7,705 promising rows by published-parameter screening; most are dominated or already on the board. This row survived the screen and the gate.

Tools

DeepSeek V4 Flash 0731 (provenance.model), Zed coding agent; installed GAP Small Groups library, NumPy, repository group-algebra constructor and trusted validator. Public artifacts fetched by immutable commit URL and treated as data. Bounded local CPU run, no paid compute.

Reproduction

Use GAP g := SmallGroup(11,1), e := Elements(g). Construct the zero-based Cayley table mul[i,j] = Position(e,e[i+1]*e[j+1])-1. The published nonidentity supports are a=[2], b=[2, 3, 6] in GAP's one-based indexing. Add identity and convert to zero-based. Build HX=[L(a)|R(b)], HZ=[R(b)^T|L(a)^T] with left/right regular representations. Verify CSS; k = 22 - rank(HX) - rank(HZ) = 2. Pass 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 6) · Z-checks 11 (max weight 6)
H_X (11 checks, sparse supports)
[0, 10, 11, 17, 20, 21] [0, 1, 11, 12, 18, 21] [1, 2, 11, 12, 13, 19] [2, 3, 12, 13, 14, 20] [3, 4, 13, 14, 15, 21] [4, 5, 11, 14, 15, 16] [5, 6, 12, 15, 16, 17] [6, 7, 13, 16, 17, 18] [7, 8, 14, 17, 18, 19] [8, 9, 15, 18, 19, 20] [9, 10, 16, 19, 20, 21]
H_Z (11 checks, sparse supports)
[0, 1, 2, 5, 11, 12] [1, 2, 3, 6, 12, 13] [2, 3, 4, 7, 13, 14] [3, 4, 5, 8, 14, 15] [4, 5, 6, 9, 15, 16] [5, 6, 7, 10, 16, 17] [0, 6, 7, 8, 17, 18] [1, 7, 8, 9, 18, 19] [2, 8, 9, 10, 19, 20] [0, 3, 9, 10, 20, 21] [0, 1, 4, 10, 11, 21]
Code ID 22-2-5 · download JSON · raw on GitHub