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[[66,4,10]] d =
n
66
k
4
d
10
kd²/n
6.061
w
6
X/Z
1
g
0.0097
r
5.0
layers
2
swaps
212

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Distance

X/Z asymmetry 1 · d_X = 10, d_Z = 10 · 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 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[7, 12, 17, 18, 29, 41, 48, 49, 50, 54]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[6, 7, 9, 13, 14, 36, 39, 43, 54, 64]
certificate exact, d = 10 · CryptoMiniSat 5.14 SAT
X: no logical < 10 exists; Z: no logical < 10 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 3 · H_Z 3
trapping sets H_X (1,3)×66 (2,2)×33 (3,3)×473 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 66 (2,2): 33 (2,4): 429 (3,3): 473 (3,5): 3531 (3,7): 396
trapping sets H_Z (1,3)×66 (2,2)×33 (3,3)×473 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 66 (2,2): 33 (2,4): 429 (3,3): 473 (3,5): 3531 (3,7): 396
witness diameter X 5.831 · Z 5.6569 (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
X checkZ checkqubit site (34)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 212 nearest-neighbor SWAPs per round in total, at most 9 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 contributed via qldpc submit
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-18
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

[[66,4,10]] — two-block group-algebra code on SmallGroup(33,1)

Direction & hypothesis

Target: unrestricted weight-6 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 = 6.06.

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_wt6wtL3_order33_k4.txt, line 1. This row is SmallGroup(33,1) with nonidentity GAP supports a=[2, 7], b=[5, 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 10, not an exact certificate. The published distance (d=10) 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

Union Alpha (maker currently anonymous), 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(33,1), 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, 7] and b=[5, 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 = 66 - 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 33 (max weight 6) · Z-checks 33 (max weight 6)
H_X (33 checks, sparse supports)
[0, 3, 31, 33, 55, 65] [0, 1, 32, 34, 57, 62] [1, 2, 6, 35, 36, 58] [1, 3, 29, 36, 53, 64] [2, 3, 4, 33, 37, 60] [4, 5, 9, 38, 39, 61] [0, 4, 6, 34, 39, 56] [5, 6, 7, 35, 40, 63] [7, 8, 12, 41, 42, 64] [2, 7, 9, 37, 42, 59] [8, 9, 10, 38, 43, 65] [10, 11, 15, 34, 44, 45] [5, 10, 12, 40, 45, 62] [11, 12, 13, 36, 41, 46] [13, 14, 18, 37, 47, 48] [8, 13, 15, 33, 43, 48] [14, 15, 16, 39, 44, 49] [16, 17, 21, 40, 50, 51] [11, 16, 18, 35, 46, 51] [17, 18, 19, 42, 47, 52] [19, 20, 24, 43, 53, 54] [14, 19, 21, 38, 49, 54] [20, 21, 22, 45, 50, 55] [22, 23, 27, 46, 56, 57] [17, 22, 24, 41, 52, 57] [23, 24, 25, 48, 53, 58] [25, 26, 30, 49, 59, 60] [20, 25, 27, 44, 55, 60] [26, 27, 28, 51, 56, 61] [28, 29, 32, 52, 62, 63] [23, 28, 30, 47, 58, 63] [29, 30, 31, 54, 59, 64] [26, 31, 32, 50, 61, 65]
H_Z (33 checks, sparse supports)
[0, 4, 15, 33, 34, 39] [1, 6, 11, 34, 35, 36] [2, 7, 18, 35, 37, 42] [2, 3, 13, 33, 36, 37] [4, 9, 14, 37, 38, 39] [5, 10, 21, 38, 40, 45] [5, 6, 16, 35, 39, 40] [7, 12, 17, 40, 41, 42] [8, 13, 24, 41, 43, 48] [8, 9, 19, 38, 42, 43] [10, 15, 20, 43, 44, 45] [11, 16, 27, 44, 46, 51] [11, 12, 22, 41, 45, 46] [13, 18, 23, 46, 47, 48] [14, 19, 30, 47, 49, 54] [14, 15, 25, 44, 48, 49] [16, 21, 26, 49, 50, 51] [17, 22, 32, 50, 52, 57] [17, 18, 28, 47, 51, 52] [19, 24, 29, 52, 53, 54] [3, 20, 25, 53, 55, 60] [20, 21, 31, 50, 54, 55] [0, 22, 27, 55, 56, 57] [6, 23, 28, 56, 58, 63] [1, 23, 24, 53, 57, 58] [2, 25, 30, 58, 59, 60] [9, 26, 31, 59, 61, 65] [4, 26, 27, 56, 60, 61] [5, 28, 32, 61, 62, 63] [1, 12, 29, 36, 62, 64] [7, 29, 30, 59, 63, 64] [3, 8, 31, 33, 64, 65] [0, 10, 32, 34, 62, 65]
Code ID 66-4-10 · download JSON · raw on GitHub