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[[66,6,10]] d =
n
66
k
6
d
10
kd²/n
9.091
w
8
X/Z
1
g
0.0066
r
6.0828
layers
2
swaps
326

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Distance

X/Z asymmetry 1 · d_X = 10, d_Z = 10 · 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 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[3, 4, 14, 29, 31, 34, 47, 54, 60, 61]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[0, 1, 7, 15, 27, 28, 34, 48, 58, 62]
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 8 · H_Z 8
qubit degrees H_X 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)
trapping sets H_X (1,2)×33 (2,2)×33 (3,2)×33 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 33 (1,6): 33 (2,2): 33 (2,4): 33 (2,6): 330 (2,8): 132 (2,10): 231 (3,2): 33 (3,4): 264 (3,6): 1947 (3,8): 2805 (3,10): 4554 (3,12): 2255 (3,14): 1287 (3,16): 33
trapping sets H_Z (1,2)×33 (2,2)×33 (3,2)×33 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 33 (1,6): 33 (2,2): 33 (2,4): 33 (2,6): 330 (2,8): 132 (2,10): 231 (3,2): 33 (3,4): 264 (3,6): 1947 (3,8): 2805 (3,10): 4554 (3,12): 2255 (3,14): 1287 (3,16): 33
witness diameter X 6.7082 · 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 = 6.083
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 326 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 2BGA (arXiv:2306.16400) on SmallGroup(33,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt8wtL2_order33_k6.txt, line 1. nonidentity supports a=[3], b=[2, 3, 11, 16, 33].
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

[[66,6,10]] — two-block group-algebra code on SmallGroup(33,1)

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 = 9.09.

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_wt8wtL2_order33_k6.txt, line 1. This row is SmallGroup(33,1) with nonidentity GAP supports a=[3], b=[2, 3, 11, 16, 33]. 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=6), 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 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(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=[3], b=[2, 3, 11, 16, 33] 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) = 6. 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 8) · Z-checks 33 (max weight 8)
H_X (33 checks, sparse supports)
[0, 29, 33, 36, 37, 55, 60, 62] [1, 31, 33, 34, 39, 56, 57, 64] [0, 2, 33, 35, 39, 40, 58, 63] [3, 32, 34, 35, 36, 53, 58, 65] [1, 4, 34, 35, 37, 42, 59, 60] [2, 5, 35, 38, 42, 43, 61, 65] [3, 6, 36, 37, 38, 39, 56, 61] [4, 7, 37, 38, 40, 45, 62, 63] [5, 8, 36, 38, 41, 45, 46, 64] [6, 9, 39, 40, 41, 42, 59, 64] [7, 10, 33, 40, 41, 43, 48, 65] [8, 11, 34, 39, 41, 44, 48, 49] [9, 12, 34, 42, 43, 44, 45, 62] [10, 13, 35, 36, 43, 44, 46, 51] [11, 14, 37, 42, 44, 47, 51, 52] [12, 15, 33, 37, 45, 46, 47, 48] [13, 16, 38, 39, 46, 47, 49, 54] [14, 17, 40, 45, 47, 50, 54, 55] [15, 18, 35, 40, 48, 49, 50, 51] [16, 19, 41, 42, 49, 50, 52, 57] [17, 20, 43, 48, 50, 53, 57, 58] [18, 21, 38, 43, 51, 52, 53, 54] [19, 22, 44, 45, 52, 53, 55, 60] [20, 23, 46, 51, 53, 56, 60, 61] [21, 24, 41, 46, 54, 55, 56, 57] [22, 25, 47, 48, 55, 56, 58, 63] [23, 26, 49, 54, 56, 59, 63, 64] [24, 27, 44, 49, 57, 58, 59, 60] [25, 28, 50, 51, 58, 59, 61, 65] [26, 29, 34, 52, 57, 59, 62, 65] [27, 30, 47, 52, 60, 61, 62, 63] [28, 31, 36, 53, 54, 61, 62, 64] [30, 32, 33, 50, 55, 63, 64, 65]
H_Z (33 checks, sparse supports)
[0, 1, 2, 10, 15, 32, 33, 35] [1, 3, 4, 11, 12, 29, 34, 37] [2, 3, 4, 5, 13, 18, 35, 38] [0, 3, 6, 8, 13, 31, 36, 39] [0, 4, 6, 7, 14, 15, 37, 40] [5, 6, 7, 8, 16, 21, 38, 41] [1, 2, 6, 9, 11, 16, 39, 42] [2, 7, 9, 10, 17, 18, 40, 43] [8, 9, 10, 11, 19, 24, 41, 44] [4, 5, 9, 12, 14, 19, 42, 45] [5, 10, 12, 13, 20, 21, 43, 46] [11, 12, 13, 14, 22, 27, 44, 47] [7, 8, 12, 15, 17, 22, 45, 48] [8, 13, 15, 16, 23, 24, 46, 49] [14, 15, 16, 17, 25, 30, 47, 50] [10, 11, 15, 18, 20, 25, 48, 51] [11, 16, 18, 19, 26, 27, 49, 52] [17, 18, 19, 20, 28, 32, 50, 53] [13, 14, 18, 21, 23, 28, 51, 54] [14, 19, 21, 22, 29, 30, 52, 55] [3, 20, 21, 22, 23, 31, 53, 56] [16, 17, 21, 24, 26, 31, 54, 57] [0, 17, 22, 24, 25, 32, 55, 58] [1, 6, 23, 24, 25, 26, 56, 59] [1, 19, 20, 24, 27, 29, 57, 60] [2, 3, 20, 25, 27, 28, 58, 61] [4, 9, 26, 27, 28, 29, 59, 62] [0, 4, 22, 23, 27, 30, 60, 63] [5, 6, 23, 28, 30, 31, 61, 64] [0, 7, 12, 29, 30, 31, 33, 62] [2, 7, 25, 26, 30, 32, 63, 65] [1, 8, 9, 26, 31, 32, 34, 64] [3, 5, 10, 28, 29, 32, 36, 65]
Code ID 66-6-10 · download JSON · raw on GitHub