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[[112,4,13]] d ≤
n
112
k
4
d
13
kd²/n
6.036
w
6
X/Z
1
g
0.0047
r
6.0
layers
2
swaps
523

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Distance

X/Z asymmetry 1 · d_X ≤ 13, d_Z ≤ 13 · 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 13 · witness weight 13 (claimed upper_bound)
witness operator (support, 13 qubits)
[24, 47, 59, 60, 64, 65, 70, 80, 81, 86, 87, 98, 103]
d_Z 13 · witness weight 13 (claimed upper_bound)
witness operator (support, 13 qubits)
[1, 4, 6, 8, 15, 25, 34, 42, 53, 59, 66, 79, 88]
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 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)
trapping sets H_X (1,2)×56 (2,2)×56 (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): 56 (1,4): 56 (2,2): 56 (2,4): 448 (2,6): 336 (3,2): 56 (3,4): 1848 (3,6): 5320 (3,8): 2688 (3,10): 224
trapping sets H_Z (1,2)×56 (2,2)×56 (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): 56 (1,4): 56 (2,2): 56 (2,4): 448 (2,6): 336 (3,2): 56 (3,4): 1848 (3,6): 5320 (3,8): 2688 (3,10): 224
witness diameter X 8.6023 · 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
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 523 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

[[112,4,13]] — two-block group-algebra code on SmallGroup(56,10)

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 = 6.04.

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_wt6wtL2_order56_k4.txt, line 43. This row is SmallGroup(56,10) with nonidentity GAP supports a=[12], b=[2, 17, 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 13, not an exact certificate. The published distance (d=13) 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(56,10), 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=[12] and b=[2, 17, 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 = 112 - rank(HX) - rank(HZ) = 4. 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 6) · Z-checks 56 (max weight 6)
H_X (56 checks, sparse supports)
[0, 50, 56, 63, 74, 105] [1, 53, 56, 57, 78, 108] [2, 54, 58, 61, 80, 90] [3, 4, 59, 71, 82, 110] [4, 42, 57, 60, 83, 96] [5, 55, 61, 65, 84, 94] [6, 7, 59, 62, 86, 111] [7, 46, 60, 63, 87, 100] [8, 9, 64, 68, 88, 98] [9, 48, 65, 69, 89, 99] [10, 11, 65, 66, 79, 90] [0, 11, 62, 67, 91, 104] [12, 13, 68, 73, 92, 102] [13, 51, 58, 69, 93, 103] [14, 15, 66, 69, 70, 94] [1, 15, 67, 71, 95, 107] [16, 17, 56, 72, 76, 96] [2, 17, 73, 77, 97, 106] [18, 19, 73, 74, 87, 98] [3, 19, 58, 70, 75, 99] [20, 21, 57, 76, 81, 100] [5, 21, 64, 77, 101, 109] [22, 23, 74, 77, 78, 102] [6, 23, 61, 75, 79, 103] [24, 25, 59, 80, 84, 104] [8, 25, 60, 81, 85, 105] [26, 27, 56, 81, 82, 95] [10, 27, 64, 78, 83, 106] [28, 29, 62, 84, 89, 107] [12, 29, 63, 72, 85, 108] [30, 31, 57, 82, 85, 86] [14, 31, 68, 83, 87, 109] [32, 33, 58, 66, 88, 92] [16, 33, 67, 89, 93, 110] [34, 35, 59, 89, 90, 103] [18, 35, 60, 72, 86, 91] [36, 37, 61, 70, 92, 97] [20, 37, 71, 80, 93, 111] [38, 39, 62, 90, 93, 94] [22, 39, 63, 76, 91, 95] [40, 41, 64, 74, 96, 100] [24, 41, 65, 75, 97, 101] [42, 43, 66, 97, 98, 109] [26, 43, 67, 80, 94, 99] [44, 45, 68, 78, 100, 105] [28, 45, 69, 79, 88, 101] [46, 47, 70, 98, 101, 102] [30, 47, 71, 84, 99, 103] [48, 49, 72, 82, 104, 107] [32, 49, 73, 83, 105, 108] [34, 50, 75, 88, 102, 106] [51, 52, 76, 86, 107, 110] [36, 52, 77, 87, 96, 108] [38, 53, 79, 92, 106, 109] [40, 54, 81, 91, 110, 111] [44, 55, 85, 95, 104, 111]
H_Z (56 checks, sparse supports)
[0, 1, 16, 26, 56, 67] [1, 4, 20, 30, 57, 71] [2, 13, 19, 32, 58, 73] [3, 6, 24, 34, 59, 75] [4, 7, 25, 35, 59, 60] [2, 5, 23, 36, 61, 77] [6, 11, 28, 38, 62, 79] [0, 7, 29, 39, 62, 63] [8, 21, 27, 40, 64, 81] [5, 9, 10, 41, 64, 65] [10, 14, 32, 42, 66, 83] [11, 15, 33, 43, 66, 67] [8, 12, 31, 44, 68, 85] [9, 13, 14, 45, 68, 69] [14, 19, 36, 46, 70, 87] [3, 15, 37, 47, 70, 71] [16, 29, 35, 48, 72, 89] [12, 17, 18, 49, 72, 73] [0, 18, 22, 40, 74, 91] [19, 23, 41, 50, 74, 75] [16, 20, 39, 51, 76, 93] [17, 21, 22, 52, 76, 77] [1, 22, 27, 44, 78, 95] [10, 23, 45, 53, 78, 79] [2, 24, 37, 43, 80, 97] [20, 25, 26, 54, 80, 81] [3, 26, 30, 48, 82, 99] [4, 27, 31, 49, 82, 83] [5, 24, 28, 47, 84, 101] [25, 29, 30, 55, 84, 85] [6, 30, 35, 51, 86, 103] [7, 18, 31, 52, 86, 87] [8, 32, 45, 50, 88, 105] [9, 28, 33, 34, 88, 89] [2, 10, 34, 38, 90, 106] [11, 35, 39, 54, 90, 91] [12, 32, 36, 53, 92, 108] [13, 33, 37, 38, 92, 93] [5, 14, 38, 43, 94, 109] [15, 26, 39, 55, 94, 95] [4, 16, 40, 52, 96, 110] [17, 36, 41, 42, 96, 97] [8, 18, 42, 46, 60, 98] [9, 19, 43, 47, 98, 99] [7, 20, 40, 44, 100, 111] [21, 41, 45, 46, 100, 101] [12, 22, 46, 50, 63, 102] [13, 23, 34, 47, 102, 103] [11, 24, 48, 55, 65, 104] [0, 25, 44, 49, 104, 105] [17, 27, 50, 53, 56, 106] [15, 28, 48, 51, 69, 107] [1, 29, 49, 52, 107, 108] [21, 31, 42, 53, 57, 109] [3, 33, 51, 54, 58, 110] [6, 37, 54, 55, 61, 111]
Code ID 112-4-13 · download JSON · raw on GitHub