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[[54,12,6]] d =
n
54
k
12
d
6
kd²/n
8.0
w
8
X/Z
1
g
0.0095
r
5.3852
layers
2
swaps
197

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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)
[0, 3, 9, 29, 35, 43]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[0, 6, 25, 27, 29, 42]
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 3–5 (mean 4.0) · H_Z 3–5 (mean 4.0)
trapping sets H_X (1,3)×27 (2,2)×27 (3,3)×108 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 27 (1,5): 27 (2,2): 27 (2,4): 81 (2,6): 324 (2,8): 216 (3,3): 108 (3,5): 1080 (3,7): 3159 (3,9): 4086 (3,11): 1647 (3,13): 135
trapping sets H_Z (1,3)×27 (2,2)×27 (3,3)×108 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 27 (1,5): 27 (2,2): 27 (2,4): 81 (2,6): 324 (2,8): 216 (3,3): 108 (3,5): 1080 (3,7): 3159 (3,9): 4086 (3,11): 1647 (3,13): 135
witness diameter X 3.6056 · Z 3.6056 (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 = 5.385
X checkZ checkqubit site (27)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 197 nearest-neighbor SWAPs per round in total, at most 8 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(27,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt8wtL3_order27_k12.txt, line 1. nonidentity supports a=[3, 8], b=[2, 3, 7, 16].
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 ≤ 8 (computed)

How this code was found

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

[[54,12,6]] — two-block group-algebra code on SmallGroup(27,1)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 8.00.

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_wt8wtL3_order27_k12.txt, line 1. This row is SmallGroup(27,1) with nonidentity GAP supports a=[3, 8], b=[2, 3, 7, 16]. 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=12), 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 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(27,1), 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=[3, 8], b=[2, 3, 7, 16] 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 = 54 - rank(HX) - rank(HZ) = 12. 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 27 (max weight 8) · Z-checks 27 (max weight 8)
H_X (27 checks, sparse supports)
[0, 16, 22, 27, 35, 49, 50, 53] [1, 21, 25, 27, 28, 36, 40, 52] [0, 2, 22, 27, 29, 31, 42, 46] [2, 3, 7, 30, 34, 43, 44, 53] [4, 24, 26, 28, 31, 41, 45, 53] [1, 5, 25, 28, 29, 32, 43, 47] [5, 6, 12, 27, 30, 33, 39, 48] [0, 2, 7, 29, 34, 36, 37, 51] [3, 7, 8, 30, 31, 35, 38, 49] [8, 9, 15, 29, 36, 42, 44, 50] [4, 10, 26, 31, 32, 37, 48, 50] [10, 11, 17, 28, 33, 38, 44, 51] [1, 5, 12, 32, 34, 39, 41, 49] [6, 12, 13, 29, 33, 35, 40, 52] [13, 14, 20, 30, 32, 36, 41, 47] [3, 8, 15, 27, 35, 37, 42, 45] [9, 15, 16, 34, 36, 38, 43, 46] [4, 10, 17, 37, 39, 44, 46, 52] [11, 17, 18, 32, 38, 40, 45, 53] [18, 19, 23, 33, 37, 41, 46, 50] [6, 13, 20, 28, 34, 40, 42, 47] [14, 20, 21, 35, 39, 41, 43, 48] [9, 16, 22, 30, 43, 45, 49, 51] [11, 18, 23, 31, 39, 45, 47, 50] [19, 23, 24, 40, 44, 46, 48, 51] [14, 21, 25, 33, 42, 48, 49, 52] [19, 24, 26, 38, 47, 51, 52, 53]
H_Z (27 checks, sparse supports)
[0, 1, 2, 6, 15, 27, 29, 34] [1, 4, 5, 11, 20, 28, 32, 39] [2, 5, 7, 9, 13, 29, 30, 34] [3, 6, 8, 14, 22, 30, 35, 42] [2, 4, 8, 10, 23, 31, 37, 44] [5, 10, 12, 14, 18, 32, 33, 39] [6, 11, 13, 19, 25, 33, 40, 47] [3, 7, 12, 16, 20, 30, 34, 35] [0, 8, 13, 15, 21, 35, 36, 42] [1, 7, 9, 14, 16, 36, 43, 49] [7, 10, 15, 17, 19, 37, 38, 44] [8, 11, 16, 18, 26, 38, 45, 50] [6, 12, 17, 21, 23, 33, 39, 40] [1, 13, 18, 20, 24, 40, 41, 47] [4, 12, 14, 19, 21, 41, 48, 52] [2, 9, 15, 20, 25, 36, 42, 43] [3, 5, 16, 21, 22, 27, 43, 49] [3, 9, 11, 17, 24, 38, 44, 45] [4, 15, 18, 22, 23, 45, 46, 50] [2, 16, 17, 19, 24, 46, 51, 53] [5, 14, 20, 23, 26, 41, 47, 48] [6, 10, 21, 24, 25, 28, 48, 52] [0, 8, 12, 22, 25, 27, 29, 49] [0, 9, 10, 19, 23, 46, 50, 51] [7, 11, 22, 24, 26, 31, 51, 53] [1, 13, 17, 25, 26, 28, 32, 52] [0, 3, 4, 18, 26, 31, 37, 53]
Code ID 54-12-6 · download JSON · raw on GitHub