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[[56,2,10]] d =
n
56
k
2
d
10
kd²/n
3.571
w
6
X/Z
1
g
0.0104
r
6.0828
layers
1
swaps
300

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Distance

X/Z asymmetry 1 · d_X = 10, d_Z = 10 · w_X = 6, w_Z = 6 (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, 12, 25, 29, 31, 37, 38, 41, 43, 49]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[7, 8, 9, 12, 15, 19, 25, 31, 37, 46]
certificate exact, d = 10 · scipy/HiGHS MILP
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 6 · H_Z 6
qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)
trapping sets H_X (1,2)×28 (2,2)×28 (3,2)×56 (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 (1,4): 28 (2,2): 28 (2,4): 224 (2,6): 168 (3,2): 56 (3,4): 924 (3,6): 2464 (3,8): 1176 (3,10): 112
trapping sets H_Z (1,2)×28 (2,2)×28 (3,2)×56 (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 (1,4): 28 (2,2): 28 (2,4): 224 (2,6): 168 (3,2): 56 (3,4): 924 (3,6): 2464 (3,8): 1176 (3,10): 112
witness diameter X 7.2801 · 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 @mathysrennela · simulated annealing over integer grid sites (research/local2d/fold_layout.py) · 2026-09-19
r = 6.083
X checkZ checkqubit site (56)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 300 nearest-neighbor SWAPs per round in total, at most 10 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 contributed via qldpc submit
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-18
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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

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

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 2·100/56 = 3.57.

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_wt6wtL2_order28_k2.txt, line 1. This row is SmallGroup(28,2) with nonidentity GAP supports a=[5], b=[3,12,17]. 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 ≤ 10, not an exact certificate. The published distance (d=10) 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

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