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[[112,12,12]] d ≤
n
112
k
12
d
12
kd²/n
15.429
w
8
X/Z
1
g
0.0076
r
6.7082
layers
2
swaps
728

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Distance

X/Z asymmetry 1 · d_X ≤ 12, d_Z ≤ 12 · 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 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[3, 4, 28, 29, 43, 72, 76, 82, 103, 104, 109, 111]
d_Z 12 · witness weight 12 (claimed upper_bound)
witness operator (support, 12 qubits)
[6, 22, 26, 48, 59, 62, 71, 79, 103, 107, 110, 111]
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 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)×56 (2,4)×224 (3,3)×14 (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 (1,5): 56 (2,4): 224 (2,6): 756 (2,8): 504 (3,3): 14 (3,5): 1428 (3,7): 8778 (3,9): 13398 (3,11): 6342 (3,13): 392
trapping sets H_Z (1,3)×56 (2,4)×224 (3,3)×14 (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 (1,5): 56 (2,4): 224 (2,6): 756 (2,8): 504 (3,3): 14 (3,5): 1428 (3,7): 8778 (3,9): 13398 (3,11): 6342 (3,13): 392
witness diameter X 7.8102 · Z 7.2111 (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
r = 6.708
X checkZ checkqubit site (56)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 728 nearest-neighbor SWAPs per round in total, at most 12 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 ≤ 8 (computed)

How this code was found

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

[[112,12,12]] — two-block group-algebra code

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 = 12·144/112 = 15.43.

What was searched

Scraped GitHub for downloadable parity-check matrix artifacts and parameter tables. From github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL3_order56_k12.txt, line 17. This row is SmallGroup(56,9) with nonidentity GAP supports a=[7,25], b=[2,12,17,36]. 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 ≤ 12, not an exact certificate. The published distance (d=12) 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(56,9), 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=[7,25] and b=[2,12,17,36] 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 = 112 - 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 56 (max weight 8) · Z-checks 56 (max weight 8)
H_X (56 checks, sparse supports)
[0, 32, 46, 56, 57, 83, 96, 106] [1, 42, 45, 56, 57, 87, 100, 109] [2, 26, 51, 58, 69, 89, 90, 110] [1, 3, 40, 59, 60, 62, 91, 104] [4, 41, 53, 60, 63, 74, 98, 105] [5, 39, 48, 61, 65, 93, 94, 111] [0, 6, 52, 59, 62, 63, 95, 107] [7, 36, 50, 60, 63, 78, 102, 108] [5, 8, 34, 64, 65, 77, 97, 98] [9, 35, 55, 61, 65, 80, 99, 104] [6, 10, 48, 58, 66, 67, 70, 99] [7, 11, 49, 56, 67, 71, 82, 110] [2, 12, 47, 68, 69, 73, 101, 102] [13, 30, 54, 58, 69, 84, 103, 107] [3, 14, 55, 61, 66, 70, 71, 103] [4, 15, 44, 57, 67, 71, 86, 111] [12, 16, 42, 56, 72, 73, 85, 105] [13, 17, 43, 58, 68, 73, 88, 106] [2, 14, 18, 64, 74, 75, 78, 106] [15, 19, 54, 59, 65, 75, 79, 90] [8, 20, 53, 57, 76, 77, 81, 108] [9, 21, 38, 61, 64, 77, 92, 109] [10, 13, 22, 68, 74, 78, 79, 109] [11, 23, 51, 62, 69, 75, 79, 94] [0, 20, 24, 59, 80, 81, 93, 110] [21, 25, 50, 60, 64, 76, 81, 96] [8, 22, 26, 60, 72, 82, 83, 86] [9, 23, 27, 66, 73, 83, 87, 98] [7, 16, 28, 62, 84, 85, 89, 111] [17, 29, 46, 63, 68, 72, 85, 100] [18, 21, 30, 63, 76, 82, 86, 87] [5, 19, 31, 70, 77, 83, 87, 102] [3, 28, 32, 65, 66, 88, 89, 101] [4, 29, 33, 67, 72, 84, 89, 104] [16, 30, 34, 67, 80, 90, 91, 94] [17, 31, 35, 56, 74, 81, 91, 95] [15, 24, 36, 69, 70, 92, 93, 97] [1, 25, 37, 71, 76, 80, 93, 107] [26, 29, 38, 71, 84, 90, 94, 95] [12, 27, 39, 57, 78, 85, 91, 95] [10, 36, 40, 73, 74, 96, 97, 108] [11, 37, 41, 58, 75, 80, 92, 97] [24, 38, 42, 75, 88, 98, 99, 102] [25, 39, 43, 59, 82, 89, 99, 103] [23, 32, 44, 77, 78, 100, 101, 105] [6, 33, 45, 61, 79, 84, 88, 101] [34, 37, 46, 79, 92, 98, 102, 103] [20, 35, 47, 62, 86, 93, 99, 103] [18, 44, 48, 81, 82, 104, 105, 111] [19, 45, 49, 64, 83, 88, 100, 105] [33, 47, 50, 66, 90, 97, 106, 109] [31, 40, 51, 85, 86, 107, 108, 110] [14, 41, 52, 68, 87, 92, 96, 108] [28, 43, 53, 70, 94, 101, 106, 109] [27, 52, 54, 72, 91, 96, 107, 110] [22, 49, 55, 76, 95, 100, 104, 111]
H_Z (56 checks, sparse supports)
[0, 1, 11, 16, 35, 56, 62, 80] [0, 1, 15, 20, 39, 57, 59, 93] [2, 10, 13, 17, 41, 58, 68, 74] [3, 6, 19, 24, 43, 59, 70, 88] [3, 4, 7, 25, 26, 60, 71, 89] [5, 9, 14, 21, 45, 61, 64, 87] [3, 6, 23, 28, 47, 62, 66, 101] [4, 6, 7, 29, 30, 63, 67, 84] [8, 18, 21, 25, 49, 64, 76, 82] [5, 8, 9, 19, 32, 65, 77, 83] [10, 14, 27, 32, 50, 66, 78, 96] [10, 11, 15, 33, 34, 67, 79, 97] [12, 17, 22, 29, 52, 68, 72, 95] [2, 12, 13, 23, 36, 69, 73, 78] [10, 14, 31, 36, 53, 70, 74, 108] [11, 14, 15, 37, 38, 71, 75, 92] [16, 26, 29, 33, 54, 72, 84, 90] [12, 16, 17, 27, 40, 73, 85, 91] [4, 18, 22, 35, 40, 74, 86, 104] [18, 19, 23, 41, 42, 75, 87, 105] [20, 25, 30, 37, 55, 76, 80, 103] [8, 20, 21, 31, 44, 77, 81, 86] [7, 18, 22, 39, 44, 78, 82, 111] [19, 22, 23, 45, 46, 79, 83, 100] [9, 24, 34, 37, 41, 80, 92, 98] [20, 24, 25, 35, 48, 81, 93, 99] [11, 26, 30, 43, 48, 58, 82, 94] [0, 26, 27, 31, 49, 83, 95, 110] [13, 28, 33, 38, 45, 84, 88, 109] [16, 28, 29, 39, 51, 85, 89, 94] [15, 26, 30, 47, 51, 69, 86, 90] [1, 27, 30, 31, 52, 87, 91, 107] [17, 32, 42, 45, 49, 56, 88, 100] [2, 28, 32, 33, 43, 89, 101, 106] [2, 19, 34, 38, 50, 64, 90, 102] [3, 34, 35, 39, 54, 65, 91, 103] [21, 36, 41, 46, 52, 63, 92, 96] [5, 24, 36, 37, 47, 93, 97, 102] [5, 23, 34, 38, 53, 77, 94, 98] [6, 35, 38, 39, 55, 61, 95, 99] [0, 25, 40, 52, 54, 59, 96, 107] [8, 36, 40, 41, 50, 60, 97, 108] [4, 8, 27, 42, 46, 57, 72, 98] [9, 10, 42, 43, 47, 73, 99, 109] [1, 29, 44, 49, 55, 71, 100, 104] [12, 32, 44, 45, 53, 57, 101, 105] [7, 12, 31, 42, 46, 56, 85, 102] [13, 14, 43, 46, 47, 68, 103, 106] [3, 9, 33, 48, 55, 61, 66, 104] [4, 16, 44, 48, 49, 67, 105, 111] [0, 17, 18, 50, 53, 63, 81, 106] [6, 13, 37, 51, 54, 58, 79, 107] [7, 20, 40, 51, 52, 62, 108, 110] [1, 21, 22, 50, 53, 60, 76, 109] [2, 11, 24, 51, 54, 69, 75, 110] [5, 15, 28, 48, 55, 65, 70, 111]
Code ID 112-12-12 · download JSON · raw on GitHub