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[[66,26,3]] d =
n
66
k
26
d
3
kd²/n
3.545
w
8
X/Z
1
g
0.0138
r
4.0
layers
2
swaps
39

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · 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 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[50, 52, 54]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[13, 16, 46]
certificate exact, d = 3 · scipy/HiGHS MILP
X: no logical < 3 exists; Z: no logical < 3 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 4 · H_Z 4
trapping sets H_X (1,4)×66 (2,4)×264 (3,0)×55 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 66 (2,4): 264 (2,6): 396 (3,0): 55 (3,4): 1089 (3,6): 3102 (3,8): 2904
trapping sets H_Z (1,4)×66 (2,4)×264 (3,0)×55 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 66 (2,4): 264 (2,6): 396 (3,0): 55 (3,4): 1089 (3,6): 3102 (3,8): 2904
witness diameter X 1.4142 · Z 2.0 (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 = 4
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 39 nearest-neighbor SWAPs per round in total, at most 2 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_wt8wtL4_order33_k26.txt, line 1. nonidentity supports a=[2, 3, 5], b=[2, 6, 8].
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

[[66,26,3]] — 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 = 3.55.

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_wt8wtL4_order33_k26.txt, line 1. This row is SmallGroup(33,1) with nonidentity GAP supports a=[2, 3, 5], b=[2, 6, 8]. 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=26), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 3, 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=[2, 3, 5], b=[2, 6, 8] 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) = 26. 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, 3, 29, 32, 33, 36, 59, 63] [0, 1, 29, 31, 33, 34, 59, 61] [0, 2, 3, 6, 35, 39, 62, 65] [1, 3, 31, 32, 34, 36, 61, 63] [0, 1, 2, 4, 35, 37, 62, 64] [2, 5, 6, 9, 33, 36, 38, 42] [1, 3, 4, 6, 37, 39, 64, 65] [2, 4, 5, 7, 33, 34, 38, 40] [5, 8, 9, 12, 35, 39, 41, 45] [4, 6, 7, 9, 34, 36, 40, 42] [5, 7, 8, 10, 35, 37, 41, 43] [8, 11, 12, 15, 38, 42, 44, 48] [7, 9, 10, 12, 37, 39, 43, 45] [8, 10, 11, 13, 38, 40, 44, 46] [11, 14, 15, 18, 41, 45, 47, 51] [10, 12, 13, 15, 40, 42, 46, 48] [11, 13, 14, 16, 41, 43, 47, 49] [14, 17, 18, 21, 44, 48, 50, 54] [13, 15, 16, 18, 43, 45, 49, 51] [14, 16, 17, 19, 44, 46, 50, 52] [17, 20, 21, 24, 47, 51, 53, 57] [16, 18, 19, 21, 46, 48, 52, 54] [17, 19, 20, 22, 47, 49, 53, 55] [20, 23, 24, 27, 50, 54, 56, 60] [19, 21, 22, 24, 49, 51, 55, 57] [20, 22, 23, 25, 50, 52, 56, 58] [23, 26, 27, 30, 53, 57, 59, 63] [22, 24, 25, 27, 52, 54, 58, 60] [23, 25, 26, 28, 53, 55, 59, 61] [26, 29, 30, 32, 56, 60, 62, 65] [25, 27, 28, 30, 55, 57, 61, 63] [26, 28, 29, 31, 56, 58, 62, 64] [28, 30, 31, 32, 58, 60, 64, 65]
H_Z (33 checks, sparse supports)
[0, 1, 5, 7, 33, 34, 35, 37] [1, 3, 7, 9, 34, 36, 37, 39] [2, 4, 8, 10, 35, 37, 38, 40] [0, 3, 5, 9, 33, 35, 36, 39] [4, 6, 10, 12, 37, 39, 40, 42] [5, 7, 11, 13, 38, 40, 41, 43] [2, 6, 8, 12, 35, 38, 39, 42] [7, 9, 13, 15, 40, 42, 43, 45] [8, 10, 14, 16, 41, 43, 44, 46] [5, 9, 11, 15, 38, 41, 42, 45] [10, 12, 16, 18, 43, 45, 46, 48] [11, 13, 17, 19, 44, 46, 47, 49] [8, 12, 14, 18, 41, 44, 45, 48] [13, 15, 19, 21, 46, 48, 49, 51] [14, 16, 20, 22, 47, 49, 50, 52] [11, 15, 17, 21, 44, 47, 48, 51] [16, 18, 22, 24, 49, 51, 52, 54] [17, 19, 23, 25, 50, 52, 53, 55] [14, 18, 20, 24, 47, 50, 51, 54] [19, 21, 25, 27, 52, 54, 55, 57] [20, 22, 26, 28, 53, 55, 56, 58] [17, 21, 23, 27, 50, 53, 54, 57] [22, 24, 28, 30, 55, 57, 58, 60] [23, 25, 29, 31, 56, 58, 59, 61] [20, 24, 26, 30, 53, 56, 57, 60] [25, 27, 31, 32, 58, 60, 61, 63] [0, 1, 26, 28, 59, 61, 62, 64] [23, 27, 29, 32, 56, 59, 60, 63] [1, 3, 28, 30, 61, 63, 64, 65] [2, 4, 29, 31, 33, 34, 62, 64] [0, 3, 26, 30, 59, 62, 63, 65] [4, 6, 31, 32, 34, 36, 64, 65] [2, 6, 29, 32, 33, 36, 62, 65]
Code ID 66-26-3 · download JSON · raw on GitHub