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[[56,4,10]] d =
n
56
k
4
d
10
kd²/n
7.143
w
8
X/Z
1
g
0.0052
r
6.0828
layers
2
swaps
220

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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)
[1, 3, 4, 7, 25, 35, 37, 48, 49, 53]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[1, 7, 10, 12, 31, 40, 45, 46, 53, 55]
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)×28 (2,2)×28 (3,2)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 28 (1,6): 28 (2,2): 28 (2,6): 336 (2,8): 168 (2,10): 84 (3,2): 28 (3,4): 56 (3,6): 2128 (3,8): 2716 (3,10): 3500 (3,12): 1232 (3,14): 56 (3,16): 28
trapping sets H_Z (1,2)×28 (2,2)×28 (3,2)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 28 (1,6): 28 (2,2): 28 (2,6): 336 (2,8): 168 (2,10): 84 (3,2): 28 (3,4): 56 (3,6): 2128 (3,8): 2716 (3,10): 3500 (3,12): 1232 (3,14): 56 (3,16): 28
witness diameter X 6.0828 · Z 6.7082 (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 (28)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 220 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(28,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt8wtL2_order28_k4.txt, line 1. nonidentity supports a=[8], b=[2, 3, 7, 15, 22].
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

[[56,4,10]] — two-block group-algebra code on SmallGroup(28,2)

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 = 7.14.

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_order28_k4.txt, line 1. This row is SmallGroup(28,2) with nonidentity GAP supports a=[8], b=[2, 3, 7, 15, 22]. 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 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(28,2), 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=[8], b=[2, 3, 7, 15, 22] 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 = 56 - 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 28 (max weight 8) · Z-checks 28 (max weight 8)
H_X (28 checks, sparse supports)
[0, 26, 28, 33, 38, 40, 46, 50] [1, 27, 28, 29, 40, 43, 48, 52] [2, 3, 28, 30, 37, 42, 44, 50] [3, 22, 29, 31, 43, 45, 51, 54] [4, 5, 29, 30, 32, 44, 47, 52] [5, 24, 31, 33, 38, 45, 53, 55] [6, 7, 28, 30, 34, 41, 46, 48] [0, 7, 31, 32, 35, 47, 49, 54] [8, 9, 29, 32, 34, 36, 48, 51] [1, 9, 33, 35, 37, 42, 49, 55] [10, 11, 30, 34, 38, 45, 50, 52] [2, 11, 31, 35, 36, 39, 51, 53] [12, 13, 32, 36, 38, 40, 52, 54] [4, 13, 33, 37, 39, 41, 46, 53] [14, 15, 28, 29, 34, 38, 42, 49] [6, 15, 35, 39, 40, 43, 54, 55] [16, 17, 29, 31, 36, 40, 42, 44] [8, 17, 37, 41, 43, 45, 50, 55] [18, 19, 30, 32, 38, 42, 46, 53] [10, 19, 31, 33, 39, 43, 44, 47] [20, 21, 32, 35, 40, 44, 46, 48] [12, 21, 28, 33, 41, 45, 47, 49] [22, 23, 34, 36, 42, 46, 50, 55] [14, 23, 35, 37, 43, 47, 48, 51] [24, 25, 36, 39, 44, 48, 50, 52] [16, 25, 30, 37, 45, 49, 51, 53] [18, 26, 39, 41, 47, 51, 52, 54] [20, 27, 34, 41, 49, 53, 54, 55]
H_Z (28 checks, sparse supports)
[0, 1, 2, 6, 14, 21, 28, 35] [1, 3, 4, 8, 14, 16, 29, 37] [2, 4, 6, 10, 18, 25, 30, 39] [3, 5, 7, 11, 16, 19, 30, 31] [4, 7, 8, 12, 18, 20, 32, 41] [0, 5, 9, 13, 19, 21, 32, 33] [6, 8, 10, 14, 22, 27, 34, 43] [7, 9, 11, 15, 20, 23, 34, 35] [8, 11, 12, 16, 22, 24, 36, 45] [2, 9, 13, 17, 23, 25, 36, 37] [0, 5, 10, 12, 14, 18, 38, 47] [11, 13, 15, 19, 24, 26, 38, 39] [0, 1, 12, 15, 16, 20, 40, 49] [6, 13, 17, 21, 26, 27, 40, 41] [2, 9, 14, 16, 18, 22, 42, 51] [1, 3, 15, 17, 19, 23, 42, 43] [2, 4, 16, 19, 20, 24, 44, 53] [3, 5, 10, 17, 21, 25, 44, 45] [0, 6, 13, 18, 20, 22, 46, 54] [4, 7, 19, 21, 23, 26, 46, 47] [1, 6, 8, 20, 23, 24, 48, 55] [7, 9, 14, 21, 25, 27, 48, 49] [0, 2, 10, 17, 22, 24, 31, 50] [3, 8, 11, 23, 25, 26, 50, 51] [1, 4, 10, 12, 24, 26, 33, 52] [5, 11, 13, 18, 25, 27, 52, 53] [3, 7, 12, 15, 26, 27, 28, 54] [5, 9, 15, 17, 22, 27, 29, 55]
Code ID 56-4-10 · download JSON · raw on GitHub