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[[24,2,6]] d =
n
24
k
2
d
6
kd²/n
3.0
w
6
X/Z
1
g
0.03
r
3.1623
layers
2
swaps
31

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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)
[0, 10, 13, 16, 18, 23]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[1, 2, 5, 6, 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 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)×96 (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): 90 (2,6): 36 (3,2): 96 (3,4): 456 (3,6): 448 (3,8): 96
trapping sets H_Z (1,2)×12 (2,2)×24 (3,2)×96 (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): 90 (2,6): 36 (3,2): 96 (3,4): 456 (3,6): 448 (3,8): 96
witness diameter X 3.6056 · Z 3.0 (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
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 31 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(12,5), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt6wtL2_order12_k2.txt, line 2. nonidentity supports a=[6], b=[2, 4, 11].
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

[[24,2,6]] — two-block group-algebra code on SmallGroup(12,5)

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 = 3.00.

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_order12_k2.txt, line 2. This row is SmallGroup(12,5) with nonidentity GAP supports a=[6], b=[2, 4, 11]. 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 ≤ 6, not an exact certificate. The published distance (d=6) 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(12,5), 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=[6], b=[2, 4, 11] 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 = 24 - 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 12 (max weight 6) · Z-checks 12 (max weight 6)
H_X (12 checks, sparse supports)
[0, 9, 12, 13, 18, 19] [1, 7, 12, 13, 20, 21] [2, 11, 14, 15, 16, 22] [1, 3, 12, 15, 17, 22] [4, 10, 14, 16, 17, 23] [0, 5, 13, 15, 17, 23] [4, 6, 14, 18, 19, 20] [5, 7, 14, 15, 19, 21] [2, 8, 16, 18, 20, 21] [3, 9, 16, 17, 19, 21] [8, 10, 12, 18, 22, 23] [6, 11, 13, 20, 22, 23]
H_Z (12 checks, sparse supports)
[0, 1, 3, 10, 12, 17] [0, 1, 5, 11, 13, 15] [2, 4, 6, 7, 14, 20] [2, 3, 5, 7, 15, 21] [2, 4, 8, 9, 16, 18] [3, 4, 5, 9, 17, 19] [0, 6, 8, 10, 18, 23] [0, 6, 7, 9, 13, 19] [1, 6, 8, 11, 20, 22] [1, 7, 8, 9, 12, 21] [2, 3, 10, 11, 16, 22] [4, 5, 10, 11, 14, 23]
Code ID 24-2-6 · download JSON · raw on GitHub