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[[56,12,8]] d =
n
56
k
12
d
8
kd²/n
13.714
w
8
X/Z
1
g
0.0195
r
7.2801
layers
1
swaps
497

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Distance

X/Z asymmetry 1 · d_X = 8, d_Z = 8 · 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 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[7, 9, 14, 23, 24, 29, 31, 47]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[6, 16, 33, 38, 39, 44, 47, 49]
certificate exact, d = 8 · scipy/HiGHS MILP
X: no logical < 8 exists; Z: no logical < 8 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 3–5 (mean 4.0) · H_Z 3–5 (mean 4.0)
trapping sets H_X (1,3)×28 (2,4)×112 (3,3)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 28 (1,5): 28 (2,4): 112 (2,6): 406 (2,8): 196 (3,3): 28 (3,5): 980 (3,7): 4452 (3,9): 4872 (3,11): 1652 (3,13): 84
trapping sets H_Z (1,3)×28 (2,4)×112 (3,3)×28 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 28 (1,5): 28 (2,4): 112 (2,6): 406 (2,8): 196 (3,3): 28 (3,5): 980 (3,7): 4452 (3,9): 4872 (3,11): 1652 (3,13): 84
witness diameter X 7.6158 · Z 8.6023 (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 = 7.28
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 497 nearest-neighbor SWAPs per round in total, at most 14 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 unrestricted (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

[[56,12,8]] — two-block group-algebra code on SmallGroup(28,4)

Direction & hypothesis

Target: unrestricted weight-8 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 (upstream main 72d7db93). Efficiency kd²/n = 12·64/56 = 13.71.

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_wt8wtL3_order28_k12.txt, line 2. This row is SmallGroup(28,4) with nonidentity GAP supports a=[6,15], b=[2,4,22,27]. 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=12), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 8, not an exact certificate. The published distance (d=8) 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(28,4), 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=[6,15] and b=[2,4,22,27] 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) = 12. 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, 18, 25, 28, 29, 34, 37, 51] [1, 20, 23, 28, 29, 35, 36, 53] [2, 15, 27, 30, 31, 32, 40, 54] [1, 3, 22, 28, 31, 33, 38, 41] [4, 17, 26, 30, 32, 33, 38, 55] [0, 5, 24, 29, 31, 33, 39, 40] [4, 6, 19, 30, 34, 35, 36, 44] [5, 7, 26, 31, 35, 37, 42, 45] [2, 8, 21, 32, 34, 36, 37, 42] [3, 9, 27, 33, 35, 37, 43, 44] [8, 10, 23, 34, 38, 39, 40, 48] [2, 9, 11, 35, 39, 41, 46, 49] [6, 12, 25, 36, 38, 40, 41, 46] [4, 7, 13, 37, 39, 41, 47, 48] [0, 12, 14, 38, 42, 43, 44, 52] [6, 13, 15, 39, 43, 45, 50, 53] [1, 10, 16, 40, 42, 44, 45, 50] [8, 11, 17, 41, 43, 45, 51, 52] [3, 16, 18, 42, 46, 47, 48, 55] [10, 17, 19, 29, 43, 47, 49, 54] [5, 14, 20, 44, 46, 48, 49, 54] [12, 15, 21, 28, 45, 47, 49, 55] [7, 20, 22, 32, 46, 50, 51, 52] [14, 21, 23, 30, 33, 47, 51, 53] [9, 18, 24, 30, 48, 50, 52, 53] [16, 19, 25, 31, 32, 49, 51, 53] [11, 24, 26, 28, 36, 50, 54, 55] [13, 22, 27, 29, 34, 52, 54, 55]
H_Z (28 checks, sparse supports)
[0, 1, 3, 21, 26, 28, 33, 42] [0, 1, 5, 19, 27, 29, 31, 44] [2, 4, 6, 23, 24, 30, 36, 39] [2, 3, 5, 7, 25, 31, 37, 46] [2, 4, 8, 22, 25, 32, 34, 41] [3, 4, 5, 9, 23, 33, 35, 48] [0, 6, 8, 10, 27, 34, 40, 43] [1, 6, 7, 9, 11, 35, 41, 50] [1, 6, 8, 12, 26, 36, 38, 45] [0, 7, 8, 9, 13, 37, 39, 52] [3, 4, 10, 12, 14, 38, 44, 47] [5, 10, 11, 13, 15, 39, 45, 54] [2, 5, 10, 12, 16, 40, 42, 49] [3, 11, 12, 13, 17, 41, 43, 55] [7, 8, 14, 16, 18, 42, 48, 51] [9, 14, 15, 17, 19, 30, 43, 49] [6, 9, 14, 16, 20, 44, 46, 53] [7, 15, 16, 17, 21, 32, 45, 47] [11, 12, 18, 20, 22, 28, 46, 52] [13, 18, 19, 21, 23, 34, 47, 53] [10, 13, 18, 20, 24, 29, 48, 50] [11, 19, 20, 21, 25, 36, 49, 51] [15, 16, 22, 24, 26, 31, 50, 55] [0, 17, 22, 23, 25, 29, 38, 51] [14, 17, 22, 24, 27, 33, 52, 54] [1, 15, 23, 24, 25, 28, 40, 53] [2, 19, 20, 26, 27, 32, 35, 54] [4, 18, 21, 26, 27, 30, 37, 55]
Code ID 56-12-8 · download JSON · raw on GitHub