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[[68,2,11]] d ≤
n
68
k
2
d
11
kd²/n
3.559
w
6
X/Z
1
g
0.0027
r
6.0
layers
2
swaps
312

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Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · 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 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[3, 6, 16, 24, 31, 35, 36, 40, 48, 51, 58]
d_Z 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[3, 15, 25, 26, 37, 38, 40, 49, 51, 56, 65]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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 6 · H_Z 6 (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)×34 (2,2)×34 (3,2)×34 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 34 (1,4): 34 (2,2): 34 (2,4): 272 (2,6): 204 (3,2): 34 (3,4): 1156 (3,6): 3094 (3,8): 1734 (3,10): 136
trapping sets H_Z (1,2)×34 (2,2)×34 (3,2)×34 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 34 (1,4): 34 (2,2): 34 (2,4): 272 (2,6): 204 (3,2): 34 (3,4): 1156 (3,6): 3094 (3,8): 1734 (3,10): 136
witness diameter X 7.2111 · Z 6.3246 (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. · 2026-10-02
r = 6
X checkZ checkqubit site (35)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 312 nearest-neighbor SWAPs per round in total, at most 10 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(34,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt6wtL2_order34_k2.txt, line 1. nonidentity supports a=[4], b=[6, 15, 22].
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

[[68,2,11]] — two-block group-algebra code on SmallGroup(34,2)

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 = 3.56.

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_order34_k2.txt, line 1. This row is SmallGroup(34,2) with nonidentity GAP supports a=[4], b=[6, 15, 22]. 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 ≤ 11, not an exact certificate. The published distance (d=11) 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(34,2), 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=[4], b=[6, 15, 22] 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 = 68 - 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 34 (max weight 6) · Z-checks 34 (max weight 6)
H_X (34 checks, sparse supports)
[0, 33, 34, 49, 54, 65] [1, 32, 35, 48, 55, 64] [1, 2, 36, 51, 56, 67] [0, 3, 37, 50, 57, 66] [3, 4, 35, 38, 53, 58] [2, 5, 34, 39, 52, 59] [5, 6, 37, 40, 55, 60] [4, 7, 36, 41, 54, 61] [7, 8, 39, 42, 57, 62] [6, 9, 38, 43, 56, 63] [9, 10, 41, 44, 59, 64] [8, 11, 40, 45, 58, 65] [11, 12, 43, 46, 61, 66] [10, 13, 42, 47, 60, 67] [13, 14, 34, 45, 48, 63] [12, 15, 35, 44, 49, 62] [15, 16, 36, 47, 50, 65] [14, 17, 37, 46, 51, 64] [17, 18, 38, 49, 52, 67] [16, 19, 39, 48, 53, 66] [19, 20, 35, 40, 51, 54] [18, 21, 34, 41, 50, 55] [21, 22, 37, 42, 53, 56] [20, 23, 36, 43, 52, 57] [23, 24, 39, 44, 55, 58] [22, 25, 38, 45, 54, 59] [25, 26, 41, 46, 57, 60] [24, 27, 40, 47, 56, 61] [27, 28, 43, 48, 59, 62] [26, 29, 42, 49, 58, 63] [29, 30, 45, 50, 61, 64] [28, 31, 44, 51, 60, 65] [31, 32, 47, 52, 63, 66] [30, 33, 46, 53, 62, 67]
H_Z (34 checks, sparse supports)
[0, 5, 14, 21, 34, 37] [1, 4, 15, 20, 35, 36] [2, 7, 16, 23, 36, 39] [3, 6, 17, 22, 37, 38] [4, 9, 18, 25, 38, 41] [5, 8, 19, 24, 39, 40] [6, 11, 20, 27, 40, 43] [7, 10, 21, 26, 41, 42] [8, 13, 22, 29, 42, 45] [9, 12, 23, 28, 43, 44] [10, 15, 24, 31, 44, 47] [11, 14, 25, 30, 45, 46] [12, 17, 26, 33, 46, 49] [13, 16, 27, 32, 47, 48] [1, 14, 19, 28, 48, 51] [0, 15, 18, 29, 49, 50] [3, 16, 21, 30, 50, 53] [2, 17, 20, 31, 51, 52] [5, 18, 23, 32, 52, 55] [4, 19, 22, 33, 53, 54] [0, 7, 20, 25, 54, 57] [1, 6, 21, 24, 55, 56] [2, 9, 22, 27, 56, 59] [3, 8, 23, 26, 57, 58] [4, 11, 24, 29, 58, 61] [5, 10, 25, 28, 59, 60] [6, 13, 26, 31, 60, 63] [7, 12, 27, 30, 61, 62] [8, 15, 28, 33, 62, 65] [9, 14, 29, 32, 63, 64] [1, 10, 17, 30, 64, 67] [0, 11, 16, 31, 65, 66] [3, 12, 19, 32, 35, 66] [2, 13, 18, 33, 34, 67]
Code ID 68-2-11 · download JSON · raw on GitHub