← back to the board
[[56,6,8]] d ≤
n
56
k
6
d
8
kd²/n
6.857
w
6
X/Z
1
g
0.0326
r
5.3852
layers
1
swaps
254

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 8, d_Z ≤ 8 · 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 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[8, 13, 28, 31, 40, 44, 45, 49]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[3, 7, 13, 14, 18, 32, 49, 50]
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 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 3 · H_Z 3
trapping sets H_X (1,3)×56 (2,4)×420 (3,3)×252 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 56 (2,4): 420 (3,3): 252 (3,5): 3444 (3,7): 560
trapping sets H_Z (1,3)×56 (2,4)×420 (3,3)×252 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 56 (2,4): 420 (3,3): 252 (3,5): 3444 (3,7): 560
witness diameter X 7.0711 · 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 @mathysrennela · simulated annealing over integer grid sites (research/local2d/fold_layout.py) · 2026-09-19
r = 5.385
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 254 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 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

[[56,6,8]] — 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 = 6.86.

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_wt6wtL3_order28_k6.txt, line 1. This row is SmallGroup(28,2) with nonidentity GAP supports a=[3, 13], b=[5, 18]. 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=6), 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(28,2), 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=[3, 13] and b=[5, 18] 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) = 6. 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, 21, 22, 28, 44, 55] [1, 14, 24, 29, 47, 50] [0, 2, 25, 30, 33, 48] [3, 16, 26, 31, 49, 52] [1, 4, 18, 28, 32, 51] [5, 19, 27, 33, 42, 54] [2, 6, 27, 34, 37, 52] [3, 7, 20, 29, 35, 53] [4, 8, 22, 30, 36, 54] [5, 9, 23, 31, 37, 46] [5, 6, 10, 29, 38, 41] [7, 11, 24, 32, 39, 55] [0, 8, 12, 31, 34, 40] [9, 13, 26, 35, 41, 50] [9, 10, 14, 32, 42, 45] [1, 11, 15, 33, 36, 43] [2, 12, 16, 35, 38, 44] [3, 13, 17, 28, 39, 45] [13, 14, 18, 36, 46, 49] [4, 15, 19, 37, 40, 47] [6, 16, 20, 39, 42, 48] [7, 17, 21, 30, 43, 49] [17, 18, 22, 40, 50, 53] [8, 19, 23, 41, 44, 51] [10, 20, 24, 43, 46, 52] [11, 21, 25, 34, 47, 53] [12, 23, 26, 45, 48, 54] [15, 25, 27, 38, 51, 55]
H_Z (28 checks, sparse supports)
[0, 4, 17, 28, 30, 40] [1, 7, 10, 29, 32, 43] [2, 8, 21, 30, 34, 44] [3, 9, 12, 31, 35, 45] [4, 11, 14, 32, 36, 47] [2, 5, 15, 33, 37, 38] [6, 12, 25, 34, 38, 48] [7, 13, 16, 35, 39, 49] [8, 15, 18, 36, 40, 51] [6, 9, 19, 37, 41, 42] [10, 16, 27, 38, 42, 52] [11, 17, 20, 39, 43, 53] [12, 19, 22, 40, 44, 54] [10, 13, 23, 41, 45, 46] [5, 14, 20, 29, 42, 46] [15, 21, 24, 43, 47, 55] [0, 16, 23, 31, 44, 48] [14, 17, 26, 45, 49, 50] [9, 18, 24, 32, 46, 50] [1, 19, 25, 33, 47, 51] [2, 20, 26, 35, 48, 52] [3, 18, 21, 28, 49, 53] [1, 13, 22, 28, 36, 50] [4, 23, 27, 37, 51, 54] [3, 6, 24, 29, 39, 52] [7, 22, 25, 30, 53, 55] [5, 8, 26, 31, 41, 54] [0, 11, 27, 33, 34, 55]
Code ID 56-6-8 · download JSON · raw on GitHub