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[[56,16,5]] d =
n
56
k
16
d
5
kd²/n
7.143
w
8
X/Z
1
g
0.0052
r
6.0828
layers
2
swaps
228

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Distance

X/Z asymmetry 1 · d_X = 5, d_Z = 5 · 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 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[3, 17, 24, 30, 31]
d_Z 5 · witness weight 5 (claimed upper_bound)
witness operator (support, 5 qubits)
[9, 15, 31, 46, 53]
certificate exact, d = 5 · scipy/HiGHS cutoff IP
X: no logical < 5 exists; Z: no logical < 5 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)×56 (2,4)×56 (3,4)×140 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 56 (2,4): 56 (2,6): 672 (3,4): 140 (3,6): 2688 (3,8): 8120 (3,10): 924
trapping sets H_Z (1,4)×56 (2,4)×56 (3,4)×140 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 56 (2,4): 56 (2,6): 672 (3,4): 140 (3,6): 2688 (3,8): 8120 (3,10): 924
witness diameter X 3.6056 · Z 4.2426 (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 228 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(28,4), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, abelian.zip, diswtabelian_wt8wtL4_order28_k16.txt, line 2. nonidentity supports a=[4, 10, 23], b=[7, 10, 25].
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

[[56,16,5]] — two-block group-algebra code on SmallGroup(28,4)

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_wt8wtL4_order28_k16.txt, line 2. This row is SmallGroup(28,4) with nonidentity GAP supports a=[4, 10, 23], b=[7, 10, 25]. 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=16), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 5, 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(28,4), 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=[4, 10, 23], b=[7, 10, 25] 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) = 16. 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, 10, 21, 23, 28, 40, 49, 54] [1, 12, 19, 25, 29, 38, 47, 55] [2, 7, 24, 26, 30, 37, 51, 52] [0, 3, 14, 25, 30, 31, 44, 53] [4, 9, 22, 27, 32, 35, 50, 53] [1, 5, 16, 23, 32, 33, 42, 51] [2, 6, 11, 27, 28, 34, 41, 55] [1, 3, 7, 18, 29, 34, 35, 48] [4, 8, 13, 26, 29, 36, 39, 54] [0, 5, 9, 20, 28, 36, 37, 46] [4, 6, 10, 15, 31, 32, 38, 45] [5, 7, 11, 22, 33, 38, 39, 52] [2, 8, 12, 17, 30, 33, 40, 43] [3, 9, 13, 24, 31, 40, 41, 50] [8, 10, 14, 19, 35, 36, 42, 49] [9, 11, 15, 26, 37, 42, 43, 55] [6, 12, 16, 21, 34, 37, 44, 47] [7, 13, 17, 27, 35, 44, 45, 54] [12, 14, 18, 23, 39, 40, 46, 53] [2, 13, 15, 19, 32, 41, 46, 47] [10, 16, 20, 25, 38, 41, 48, 51] [4, 11, 17, 21, 30, 39, 48, 49] [0, 16, 18, 22, 29, 43, 44, 50] [6, 17, 19, 23, 36, 45, 50, 51] [1, 14, 20, 24, 28, 42, 45, 52] [8, 15, 21, 25, 34, 43, 52, 53] [3, 20, 22, 26, 33, 47, 48, 54] [5, 18, 24, 27, 31, 46, 49, 55]
H_Z (28 checks, sparse supports)
[0, 6, 9, 24, 28, 31, 37, 50] [1, 7, 8, 22, 29, 33, 35, 52] [2, 3, 12, 21, 30, 34, 40, 47] [3, 10, 13, 27, 31, 35, 41, 54] [4, 5, 10, 19, 32, 36, 38, 49] [5, 11, 12, 26, 33, 37, 39, 55] [6, 7, 16, 25, 34, 38, 44, 51] [4, 7, 14, 17, 30, 35, 39, 45] [8, 9, 14, 23, 36, 40, 42, 53] [2, 9, 15, 16, 32, 37, 41, 43] [1, 10, 11, 20, 28, 38, 42, 48] [8, 11, 18, 21, 34, 39, 43, 49] [0, 12, 13, 18, 29, 40, 44, 46] [6, 13, 19, 20, 36, 41, 45, 47] [5, 14, 15, 24, 31, 42, 46, 52] [12, 15, 22, 25, 38, 43, 47, 53] [3, 16, 17, 22, 33, 44, 48, 50] [10, 17, 23, 24, 40, 45, 49, 51] [9, 18, 19, 27, 35, 46, 50, 55] [1, 16, 19, 26, 29, 42, 47, 51] [7, 20, 21, 26, 37, 48, 52, 54] [0, 14, 21, 27, 28, 44, 49, 53] [4, 13, 22, 23, 32, 39, 50, 54] [2, 5, 20, 23, 28, 33, 46, 51] [2, 11, 24, 25, 30, 41, 52, 55] [3, 4, 18, 25, 29, 31, 48, 53] [0, 8, 17, 26, 30, 36, 43, 54] [1, 6, 15, 27, 32, 34, 45, 55]
Code ID 56-16-5 · download JSON · raw on GitHub