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[[64,2,12]] d ≤
n
64
k
2
d
12
kd²/n
4.5
w
8
X/Z
1
g
0.0035
r
6.0
layers
2
swaps
240

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Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[3, 4, 5, 13, 17, 21, 25, 31, 34, 35, 44, 58]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[3, 7, 12, 21, 22, 29, 37, 46, 49, 51, 58, 62]
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)×32 (2,2)×32 (3,2)×32 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 32 (1,6): 32 (2,2): 32 (2,4): 32 (2,6): 352 (2,8): 112 (2,10): 160 (3,2): 32 (3,4): 288 (3,6): 2176 (3,8): 2624 (3,10): 3776 (3,12): 1152 (3,14): 800
trapping sets H_Z (1,2)×32 (2,2)×32 (3,2)×32 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 32 (1,6): 32 (2,2): 32 (2,4): 32 (2,6): 352 (2,8): 112 (2,10): 160 (3,2): 32 (3,4): 288 (3,6): 2176 (3,8): 2624 (3,10): 3776 (3,12): 1152 (3,14): 800
witness diameter X 7.2801 · Z 7.6158 (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
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 240 nearest-neighbor SWAPs per round in total, at most 8 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(32,16), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt8wtL2_order32_k2.txt, line 3. nonidentity supports a=[2], b=[2, 8, 11, 15, 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

[[64,2,12]] — two-block group-algebra code on SmallGroup(32,16)

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 = 4.50.

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_order32_k2.txt, line 3. This row is SmallGroup(32,16) with nonidentity GAP supports a=[2], b=[2, 8, 11, 15, 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 ≤ 12, 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(32,16), 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, 8, 11, 15, 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 = 64 - 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 32 (max weight 8) · Z-checks 32 (max weight 8)
H_X (32 checks, sparse supports)
[0, 29, 32, 41, 45, 53, 61, 62] [0, 1, 32, 33, 46, 51, 57, 63] [2, 31, 34, 50, 54, 57, 60, 63] [1, 3, 33, 34, 35, 37, 52, 61] [4, 7, 33, 36, 39, 42, 46, 53] [5, 19, 33, 37, 40, 51, 54, 57] [2, 6, 34, 38, 55, 58, 61, 62] [3, 7, 32, 35, 38, 39, 41, 47] [4, 8, 35, 36, 40, 48, 52, 57] [5, 9, 35, 37, 41, 45, 58, 61] [6, 10, 32, 38, 42, 44, 59, 63] [11, 16, 35, 38, 43, 48, 55, 60] [12, 26, 38, 44, 45, 49, 58, 62] [8, 13, 39, 40, 43, 45, 47, 61] [9, 14, 32, 39, 41, 44, 46, 51] [15, 20, 35, 40, 41, 47, 52, 55] [10, 16, 33, 34, 42, 48, 50, 56] [11, 17, 39, 42, 43, 49, 59, 62] [12, 18, 42, 44, 50, 51, 54, 63] [13, 19, 32, 36, 45, 49, 51, 53] [14, 20, 33, 36, 37, 46, 50, 52] [15, 21, 39, 45, 46, 47, 53, 59] [17, 22, 36, 48, 49, 54, 56, 63] [18, 23, 34, 37, 48, 50, 55, 58] [24, 27, 42, 46, 49, 50, 56, 59] [21, 25, 36, 51, 52, 53, 56, 57] [22, 26, 34, 40, 43, 54, 58, 60] [23, 27, 38, 41, 43, 44, 55, 59] [24, 28, 48, 52, 54, 55, 56, 60] [25, 29, 37, 40, 47, 57, 60, 61] [28, 30, 43, 47, 58, 59, 60, 62] [30, 31, 44, 49, 53, 56, 62, 63]
H_Z (32 checks, sparse supports)
[0, 1, 7, 10, 14, 19, 32, 33] [1, 3, 4, 5, 16, 20, 33, 35] [2, 3, 6, 16, 23, 26, 34, 38] [3, 7, 8, 9, 11, 15, 35, 39] [4, 8, 19, 20, 22, 25, 36, 40] [3, 5, 9, 20, 23, 29, 37, 41] [6, 7, 10, 11, 12, 27, 38, 42] [4, 7, 13, 14, 17, 21, 36, 39] [5, 8, 13, 15, 26, 29, 40, 45] [0, 7, 9, 14, 15, 27, 41, 46] [4, 10, 16, 17, 18, 24, 42, 48] [11, 13, 17, 26, 27, 30, 43, 49] [10, 12, 14, 18, 27, 31, 44, 50] [0, 9, 12, 13, 19, 21, 45, 51] [1, 4, 14, 20, 21, 24, 46, 52] [7, 13, 15, 21, 29, 30, 47, 53] [8, 11, 16, 22, 23, 28, 43, 48] [12, 17, 19, 22, 24, 31, 49, 54] [2, 16, 18, 20, 23, 24, 50, 55] [1, 5, 14, 18, 19, 25, 37, 51] [3, 8, 15, 20, 25, 28, 47, 52] [0, 4, 19, 21, 25, 31, 53, 57] [2, 5, 18, 22, 26, 28, 54, 58] [6, 11, 15, 23, 27, 28, 55, 59] [16, 22, 24, 25, 28, 31, 56, 60] [1, 2, 5, 8, 25, 29, 57, 61] [6, 9, 12, 23, 26, 30, 44, 58] [10, 17, 21, 24, 27, 30, 56, 59] [2, 11, 26, 28, 29, 30, 60, 62] [0, 3, 6, 9, 13, 29, 32, 61] [0, 6, 12, 17, 30, 31, 62, 63] [1, 2, 10, 18, 22, 31, 34, 63]
Code ID 64-2-12 · download JSON · raw on GitHub