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[[62,2,11]] d ≤
n
62
k
2
d
11
kd²/n
3.903
w
8
X/Z
1
g
0.0029
r
6.0828
layers
2
swaps
260

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Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · 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 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[0, 3, 5, 7, 13, 15, 16, 20, 34, 35, 48]
d_Z 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[7, 15, 16, 20, 22, 29, 31, 38, 48, 51, 61]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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)×31 (2,2)×31 (3,2)×31 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 31 (1,6): 31 (2,2): 31 (2,4): 62 (2,6): 248 (2,8): 124 (2,10): 217 (3,2): 31 (3,4): 465 (3,6): 1457 (3,8): 2387 (3,10): 3782 (3,12): 1891 (3,14): 1085 (3,16): 62
trapping sets H_Z (1,2)×31 (2,2)×31 (3,2)×31 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 31 (1,6): 31 (2,2): 31 (2,4): 62 (2,6): 248 (2,8): 124 (2,10): 217 (3,2): 31 (3,4): 465 (3,6): 1457 (3,8): 2387 (3,10): 3782 (3,12): 1891 (3,14): 1085 (3,16): 62
witness diameter X 6.0 · 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 (32)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 260 nearest-neighbor SWAPs per round in total, at most 7 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(31,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt8wtL2_order31_k2.txt, line 1. nonidentity supports a=[2], b=[2, 3, 10, 17, 20].
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

[[62,2,11]] — two-block group-algebra code on SmallGroup(31,1)

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 = 3.90.

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_order31_k2.txt, line 1. This row is SmallGroup(31,1) with nonidentity GAP supports a=[2], b=[2, 3, 10, 17, 20]. 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=2), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 11, 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(31,1), 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], b=[2, 3, 10, 17, 20] 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 = 62 - rank(HX) - rank(HZ) = 2. 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 31 (max weight 8) · Z-checks 31 (max weight 8)
H_X (31 checks, sparse supports)
[0, 30, 31, 43, 46, 53, 60, 61] [0, 1, 31, 32, 44, 47, 54, 61] [1, 2, 31, 32, 33, 45, 48, 55] [2, 3, 32, 33, 34, 46, 49, 56] [3, 4, 33, 34, 35, 47, 50, 57] [4, 5, 34, 35, 36, 48, 51, 58] [5, 6, 35, 36, 37, 49, 52, 59] [6, 7, 36, 37, 38, 50, 53, 60] [7, 8, 37, 38, 39, 51, 54, 61] [8, 9, 31, 38, 39, 40, 52, 55] [9, 10, 32, 39, 40, 41, 53, 56] [10, 11, 33, 40, 41, 42, 54, 57] [11, 12, 34, 41, 42, 43, 55, 58] [12, 13, 35, 42, 43, 44, 56, 59] [13, 14, 36, 43, 44, 45, 57, 60] [14, 15, 37, 44, 45, 46, 58, 61] [15, 16, 31, 38, 45, 46, 47, 59] [16, 17, 32, 39, 46, 47, 48, 60] [17, 18, 33, 40, 47, 48, 49, 61] [18, 19, 31, 34, 41, 48, 49, 50] [19, 20, 32, 35, 42, 49, 50, 51] [20, 21, 33, 36, 43, 50, 51, 52] [21, 22, 34, 37, 44, 51, 52, 53] [22, 23, 35, 38, 45, 52, 53, 54] [23, 24, 36, 39, 46, 53, 54, 55] [24, 25, 37, 40, 47, 54, 55, 56] [25, 26, 38, 41, 48, 55, 56, 57] [26, 27, 39, 42, 49, 56, 57, 58] [27, 28, 40, 43, 50, 57, 58, 59] [28, 29, 41, 44, 51, 58, 59, 60] [29, 30, 42, 45, 52, 59, 60, 61]
H_Z (31 checks, sparse supports)
[0, 1, 2, 9, 16, 19, 31, 32] [1, 2, 3, 10, 17, 20, 32, 33] [2, 3, 4, 11, 18, 21, 33, 34] [3, 4, 5, 12, 19, 22, 34, 35] [4, 5, 6, 13, 20, 23, 35, 36] [5, 6, 7, 14, 21, 24, 36, 37] [6, 7, 8, 15, 22, 25, 37, 38] [7, 8, 9, 16, 23, 26, 38, 39] [8, 9, 10, 17, 24, 27, 39, 40] [9, 10, 11, 18, 25, 28, 40, 41] [10, 11, 12, 19, 26, 29, 41, 42] [11, 12, 13, 20, 27, 30, 42, 43] [0, 12, 13, 14, 21, 28, 43, 44] [1, 13, 14, 15, 22, 29, 44, 45] [2, 14, 15, 16, 23, 30, 45, 46] [0, 3, 15, 16, 17, 24, 46, 47] [1, 4, 16, 17, 18, 25, 47, 48] [2, 5, 17, 18, 19, 26, 48, 49] [3, 6, 18, 19, 20, 27, 49, 50] [4, 7, 19, 20, 21, 28, 50, 51] [5, 8, 20, 21, 22, 29, 51, 52] [6, 9, 21, 22, 23, 30, 52, 53] [0, 7, 10, 22, 23, 24, 53, 54] [1, 8, 11, 23, 24, 25, 54, 55] [2, 9, 12, 24, 25, 26, 55, 56] [3, 10, 13, 25, 26, 27, 56, 57] [4, 11, 14, 26, 27, 28, 57, 58] [5, 12, 15, 27, 28, 29, 58, 59] [6, 13, 16, 28, 29, 30, 59, 60] [0, 7, 14, 17, 29, 30, 60, 61] [0, 1, 8, 15, 18, 30, 31, 61]
Code ID 62-2-11 · download JSON · raw on GitHub