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[[54,4,9]] d =
n
54
k
4
d
9
kd²/n
6.0
w
7
X/Z
1
g
0.0119
r
6.7082
layers
1
swaps
386

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Distance

X/Z asymmetry 1 · d_X = 9, d_Z = 9 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[3, 9, 16, 24, 32, 35, 36, 48, 51]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[0, 1, 10, 11, 14, 27, 31, 37, 45]
certificate exact, d = 9 · scipy/HiGHS MILP
X: no logical < 9 exists; Z: no logical < 9 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 7 · H_Z 7
qubit degrees H_X 2–5 (mean 3.5) · H_Z 2–5 (mean 3.5)
trapping sets H_X (1,2)×27 (2,2)×27 (3,2)×27 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 27 (1,5): 27 (2,2): 27 (2,5): 270 (2,6): 27 (2,8): 216 (3,2): 27 (3,3): 27 (3,4): 27 (3,5): 1269 (3,6): 621 (3,7): 351 (3,8): 3159 (3,9): 603 (3,10): 432 (3,11): 1377 (3,13): 135
trapping sets H_Z (1,2)×27 (2,2)×27 (3,2)×27 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 27 (1,5): 27 (2,2): 27 (2,5): 270 (2,6): 27 (2,8): 216 (3,2): 27 (3,3): 27 (3,4): 27 (3,5): 1269 (3,6): 621 (3,7): 351 (3,8): 3159 (3,9): 603 (3,10): 432 (3,11): 1377 (3,13): 135
witness diameter X 8.0623 · Z 7.2111 (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 @mathysrennela · simulated annealing over integer grid sites (research/local2d/fold_layout.py) · 2026-09-19
r = 6.708
X checkZ checkqubit site (54)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 386 nearest-neighbor SWAPs per round in total, at most 13 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_wt7wtL2_order27_k4.txt, line 1. nonidentity supports a=[3], b=[2, 8, 16, 24].
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

[[54,4,9]] — two-block group-algebra code on SmallGroup(27,1)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 6.00.

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_wt7wtL2_order27_k4.txt, line 1. This row is SmallGroup(27,1) with nonidentity GAP supports a=[3], b=[2, 8, 16, 24]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 9, not an exact certificate. The published distance (d=9) 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], b=[2, 8, 16, 24] 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) = 4. 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 7) · Z-checks 27 (max weight 7)
H_X (27 checks, sparse supports)
[0, 22, 27, 33, 35, 43, 53] [1, 25, 27, 28, 38, 40, 48] [0, 2, 29, 31, 40, 42, 49] [3, 7, 29, 30, 41, 43, 44] [4, 26, 28, 31, 35, 45, 51] [1, 5, 29, 32, 45, 47, 52] [6, 12, 30, 32, 33, 46, 48] [2, 7, 27, 34, 36, 37, 47] [3, 8, 34, 35, 38, 48, 49] [9, 15, 28, 29, 35, 36, 50] [4, 10, 32, 37, 42, 50, 53] [11, 17, 33, 37, 38, 43, 51] [5, 12, 28, 34, 39, 41, 50] [6, 13, 35, 39, 40, 51, 52] [14, 20, 31, 32, 36, 40, 41] [8, 15, 27, 30, 42, 45, 52] [9, 16, 32, 34, 42, 43, 46] [10, 17, 31, 36, 39, 44, 46] [11, 18, 40, 44, 45, 49, 53] [19, 23, 29, 37, 41, 45, 46] [13, 20, 28, 33, 42, 47, 53] [14, 21, 37, 39, 43, 47, 48] [16, 22, 30, 36, 39, 49, 51] [18, 23, 27, 31, 38, 47, 50] [19, 24, 34, 44, 48, 50, 51] [21, 25, 33, 41, 44, 49, 52] [24, 26, 30, 38, 46, 52, 53]
H_Z (27 checks, sparse supports)
[0, 1, 7, 15, 23, 27, 29] [1, 4, 9, 12, 20, 28, 32] [2, 3, 5, 9, 19, 29, 34] [3, 6, 15, 22, 26, 30, 35] [2, 4, 14, 17, 23, 31, 37] [5, 6, 10, 14, 16, 32, 39] [0, 6, 11, 20, 25, 33, 40] [7, 8, 12, 16, 24, 30, 34] [0, 4, 8, 9, 13, 35, 42] [7, 9, 14, 17, 22, 36, 43] [7, 10, 11, 19, 21, 37, 44] [1, 8, 11, 23, 26, 38, 45] [12, 13, 17, 21, 22, 33, 39] [1, 2, 13, 14, 18, 40, 47] [3, 12, 14, 19, 25, 41, 48] [2, 10, 15, 16, 20, 36, 42] [0, 3, 11, 16, 21, 43, 49] [3, 17, 18, 24, 25, 38, 44] [4, 5, 15, 18, 19, 45, 50] [6, 16, 17, 19, 26, 46, 51] [5, 7, 20, 21, 23, 41, 47] [1, 6, 8, 21, 24, 48, 52] [2, 8, 18, 22, 25, 27, 49] [9, 10, 12, 23, 24, 46, 50] [4, 11, 13, 22, 24, 51, 53] [5, 13, 15, 25, 26, 28, 52] [0, 10, 18, 20, 26, 31, 53]
Code ID 54-4-9 · download JSON · raw on GitHub