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[[110,8,11]] d ≤
n
110
k
8
d
11
kd²/n
8.8
w
7
X/Z
1
g
0.0037
r
7.0
layers
2
swaps
720

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Distance

X/Z asymmetry 1 · d_X ≤ 11, d_Z ≤ 11 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[6, 29, 45, 50, 58, 64, 69, 79, 84, 105, 106]
d_Z 11 · witness weight 11 (claimed upper_bound)
witness operator (support, 11 qubits)
[3, 21, 22, 27, 28, 43, 47, 66, 76, 85, 98]
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 7 · H_Z 7
qubit degrees H_X 2–5 (mean 3.5) · H_Z 2–5 (mean 3.5)
trapping sets H_X (1,2)×55 (2,2)×55 (3,2)×55 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 55 (1,5): 55 (2,2): 55 (2,5): 550 (2,8): 550 (3,2): 55 (3,5): 2750 (3,6): 440 (3,7): 275 (3,8): 8855 (3,9): 825 (3,10): 1100 (3,11): 6325 (3,13): 550
trapping sets H_Z (1,2)×55 (2,2)×55 (3,2)×55 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 55 (1,5): 55 (2,2): 55 (2,5): 550 (2,8): 550 (3,2): 55 (3,5): 2750 (3,6): 440 (3,7): 275 (3,8): 8855 (3,9): 825 (3,10): 1100 (3,11): 6325 (3,13): 550
witness diameter X 8.4853 · Z 6.3246 (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 = 7
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 720 nearest-neighbor SWAPs per round in total, at most 13 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

[[110,8,11]] — two-block group-algebra code on SmallGroup(55,1)

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. Efficiency kd²/n = 8.80.

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_wt7wtL2_order55_k8.txt, line 4. This row is SmallGroup(55,1) with nonidentity GAP supports a=[3], b=[2, 8, 16, 47]. 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=8), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 11, not an exact certificate. The published distance (d=11) 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(55,1), 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] and b=[2, 8, 16, 47] 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 = 110 - rank(HX) - rank(HZ) = 8. 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 55 (max weight 7) · Z-checks 55 (max weight 7)
H_X (55 checks, sparse supports)
[0, 44, 55, 65, 71, 77, 88] [1, 33, 55, 56, 75, 81, 92] [0, 2, 56, 57, 61, 80, 102] [3, 32, 56, 58, 60, 85, 96] [4, 38, 58, 59, 64, 65, 105] [2, 5, 60, 72, 73, 95, 105] [6, 16, 58, 61, 63, 69, 100] [7, 37, 55, 61, 62, 68, 107] [8, 43, 63, 76, 77, 79, 107] [5, 9, 64, 93, 96, 97, 107] [10, 49, 61, 65, 67, 73, 84] [11, 21, 56, 65, 66, 72, 94] [12, 42, 67, 80, 81, 83, 94] [13, 48, 68, 94, 97, 100, 101] [9, 14, 59, 67, 69, 70, 86] [15, 52, 55, 58, 70, 76, 98] [16, 26, 64, 71, 85, 87, 98] [17, 47, 72, 84, 98, 101, 104] [1, 18, 57, 62, 71, 73, 90] [14, 19, 74, 76, 78, 85, 92] [20, 54, 68, 69, 75, 91, 102] [21, 31, 76, 88, 89, 102, 104] [22, 51, 59, 66, 74, 75, 77] [4, 23, 69, 78, 80, 82, 96] [19, 24, 62, 66, 79, 98, 100] [10, 25, 80, 89, 92, 93, 105] [26, 36, 60, 62, 70, 78, 81] [3, 27, 64, 73, 82, 86, 100] [8, 28, 66, 70, 83, 84, 102] [24, 29, 63, 84, 87, 108, 109] [15, 30, 57, 63, 66, 82, 85] [31, 41, 68, 77, 84, 86, 90] [7, 32, 57, 70, 87, 88, 105] [13, 33, 67, 88, 91, 99, 109] [29, 34, 58, 75, 83, 89, 101] [20, 35, 72, 74, 81, 88, 90] [36, 46, 57, 59, 91, 92, 107] [12, 37, 71, 92, 95, 99, 103] [18, 38, 60, 61, 87, 93, 104] [34, 39, 82, 90, 91, 94, 103] [25, 40, 59, 62, 94, 95, 96] [41, 50, 75, 79, 96, 103, 106] [17, 42, 63, 65, 89, 91, 97] [23, 43, 74, 86, 95, 98, 106] [39, 44, 68, 81, 99, 104, 106] [30, 45, 60, 83, 100, 106, 108] [46, 53, 55, 67, 93, 95, 101] [22, 47, 78, 79, 90, 102, 108] [28, 48, 72, 85, 89, 103, 108] [35, 49, 56, 71, 79, 97, 104] [6, 50, 74, 82, 83, 105, 109] [27, 51, 69, 76, 93, 106, 109] [40, 52, 78, 86, 87, 99, 107] [11, 53, 73, 80, 97, 99, 108] [45, 54, 64, 77, 101, 103, 109]
H_Z (55 checks, sparse supports)
[0, 1, 7, 15, 46, 55, 57] [1, 2, 3, 11, 49, 56, 73] [2, 18, 30, 32, 36, 57, 60] [3, 4, 6, 15, 34, 58, 82] [4, 14, 22, 36, 40, 59, 78] [3, 5, 26, 38, 45, 60, 64] [2, 6, 7, 10, 38, 61, 105] [7, 18, 24, 26, 40, 62, 87] [6, 8, 29, 30, 42, 63, 83] [4, 9, 16, 27, 54, 64, 69] [0, 4, 10, 11, 42, 65, 80] [11, 22, 24, 28, 30, 66, 108] [10, 12, 14, 33, 46, 67, 92] [7, 13, 20, 31, 44, 68, 88] [6, 14, 20, 23, 51, 69, 74] [14, 15, 26, 28, 32, 70, 85] [0, 16, 18, 37, 49, 61, 71] [5, 11, 17, 35, 48, 72, 97] [5, 10, 18, 27, 53, 73, 93] [19, 22, 35, 43, 50, 74, 79] [1, 20, 22, 34, 41, 75, 90] [8, 15, 19, 21, 51, 66, 76] [0, 8, 22, 31, 54, 77, 102] [19, 23, 26, 47, 52, 78, 98] [8, 24, 41, 47, 49, 79, 84] [2, 12, 23, 25, 53, 80, 95] [1, 12, 26, 35, 44, 71, 81] [23, 27, 30, 39, 50, 82, 106] [12, 28, 34, 45, 50, 83, 103] [10, 17, 28, 29, 31, 84, 89] [3, 16, 19, 30, 48, 85, 100] [14, 27, 31, 43, 52, 76, 86] [16, 29, 32, 38, 52, 58, 87] [0, 21, 32, 33, 35, 56, 88] [21, 25, 34, 42, 48, 89, 94] [18, 31, 35, 39, 47, 90, 104] [20, 33, 36, 39, 42, 81, 91] [1, 19, 25, 36, 37, 62, 92] [9, 25, 38, 46, 51, 59, 93] [11, 12, 13, 39, 40, 94, 99] [5, 37, 40, 43, 46, 95, 107] [3, 9, 23, 40, 41, 86, 96] [9, 13, 42, 49, 53, 67, 97] [15, 16, 17, 24, 43, 63, 98] [33, 37, 44, 52, 53, 55, 99] [6, 13, 24, 27, 45, 100, 109] [13, 17, 34, 46, 54, 91, 101] [2, 20, 21, 28, 47, 72, 102] [37, 39, 41, 48, 54, 68, 103] [17, 21, 38, 44, 49, 65, 104] [4, 5, 25, 32, 50, 96, 105] [41, 43, 44, 45, 51, 77, 106] [7, 8, 9, 36, 52, 70, 107] [29, 45, 47, 48, 53, 101, 108] [29, 33, 50, 51, 54, 75, 109]
Code ID 110-8-11 · download JSON · raw on GitHub