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[[116,2,15]] d ≤
n
116
k
2
d
15
kd²/n
3.879
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[12, 30, 35, 37, 69, 70, 88, 89, 95, 97, 99, 100, 101, 102, 104]
d_Z 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[7, 13, 25, 39, 41, 43, 45, 46, 47, 49, 52, 63, 81, 104, 105]
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 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)
trapping sets H_X (1,2)×58 (2,2)×58 (3,2)×58 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 58 (1,6): 58 (2,2): 58 (2,4): 58 (2,6): 580 (2,8): 203 (2,10): 464 (3,2): 58 (3,4): 522 (3,6): 3132 (3,8): 4060 (3,10): 9512 (3,12): 6496 (3,14): 2668 (3,16): 290
trapping sets H_Z (1,2)×58 (2,2)×58 (3,2)×58 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 58 (1,6): 58 (2,2): 58 (2,4): 58 (2,6): 580 (2,8): 203 (2,10): 464 (3,2): 58 (3,4): 522 (3,6): 3132 (3,8): 4060 (3,10): 9512 (3,12): 6496 (3,14): 2668 (3,16): 290

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(58,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order58_k2.txt, line 8. nonidentity supports a=[3], b=[2, 3, 7, 25, 49].
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-19
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

[[116,2,15]] — two-block group-algebra code on SmallGroup(58,1)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 3.88.

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_wt8wtL2_order58_k2.txt, line 8. This row is SmallGroup(58,1) with nonidentity GAP supports a=[3], b=[2, 3, 7, 25, 49]. 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=2), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 15, not an exact certificate. The published distance 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

DeepSeek V4 Flash 0731 (provenance.model), 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(58,1), 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], b=[2, 3, 7, 25, 49] 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 = 116 - rank(HX) - rank(HZ) = 2. 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 58 (max weight 8) · Z-checks 58 (max weight 8)
H_X (58 checks, sparse supports)
[0, 56, 58, 59, 68, 92, 110, 114] [1, 3, 58, 59, 69, 93, 111, 115] [0, 2, 58, 60, 70, 94, 112, 115] [3, 5, 59, 61, 71, 95, 113, 114] [2, 4, 60, 62, 72, 96, 113, 114] [5, 7, 61, 63, 73, 97, 112, 115] [4, 6, 58, 62, 64, 74, 98, 111] [7, 9, 59, 63, 65, 75, 99, 110] [6, 8, 60, 64, 66, 76, 100, 109] [9, 11, 61, 65, 67, 77, 101, 108] [8, 10, 62, 66, 68, 78, 102, 107] [11, 13, 63, 67, 69, 79, 103, 106] [10, 12, 64, 68, 70, 80, 104, 105] [13, 15, 65, 69, 71, 81, 104, 105] [12, 14, 66, 70, 72, 82, 103, 106] [15, 17, 67, 71, 73, 83, 102, 107] [14, 16, 68, 72, 74, 84, 101, 108] [17, 19, 69, 73, 75, 85, 100, 109] [16, 18, 70, 74, 76, 86, 99, 110] [19, 21, 71, 75, 77, 87, 98, 111] [18, 20, 72, 76, 78, 88, 97, 112] [21, 23, 73, 77, 79, 89, 96, 113] [20, 22, 74, 78, 80, 90, 95, 114] [23, 25, 75, 79, 81, 91, 94, 115] [22, 24, 58, 76, 80, 82, 92, 93] [25, 27, 59, 77, 81, 83, 92, 93] [24, 26, 60, 78, 82, 84, 91, 94] [27, 29, 61, 79, 83, 85, 90, 95] [26, 28, 62, 80, 84, 86, 89, 96] [29, 31, 63, 81, 85, 87, 88, 97] [28, 30, 64, 82, 86, 87, 88, 98] [31, 33, 65, 83, 86, 87, 89, 99] [30, 32, 66, 84, 85, 88, 90, 100] [33, 35, 67, 84, 85, 89, 91, 101] [32, 34, 68, 83, 86, 90, 92, 102] [35, 37, 69, 82, 87, 91, 93, 103] [34, 36, 70, 81, 88, 92, 94, 104] [37, 39, 71, 80, 89, 93, 95, 105] [36, 38, 72, 79, 90, 94, 96, 106] [39, 41, 73, 78, 91, 95, 97, 107] [38, 40, 74, 77, 92, 96, 98, 108] [41, 43, 75, 76, 93, 97, 99, 109] [40, 42, 75, 76, 94, 98, 100, 110] [43, 45, 74, 77, 95, 99, 101, 111] [42, 44, 73, 78, 96, 100, 102, 112] [45, 47, 72, 79, 97, 101, 103, 113] [44, 46, 71, 80, 98, 102, 104, 114] [47, 49, 70, 81, 99, 103, 105, 115] [46, 48, 58, 69, 82, 100, 104, 106] [49, 51, 59, 68, 83, 101, 105, 107] [48, 50, 60, 67, 84, 102, 106, 108] [51, 53, 61, 66, 85, 103, 107, 109] [50, 52, 62, 65, 86, 104, 108, 110] [53, 55, 63, 64, 87, 105, 109, 111] [52, 54, 63, 64, 88, 106, 110, 112] [55, 57, 62, 65, 89, 107, 111, 113] [54, 56, 61, 66, 90, 108, 112, 114] [1, 57, 60, 67, 91, 109, 113, 115]
H_Z (58 checks, sparse supports)
[0, 1, 2, 6, 24, 48, 58, 60] [0, 1, 3, 7, 25, 49, 59, 115] [2, 4, 8, 26, 50, 57, 60, 62] [3, 5, 9, 27, 51, 56, 59, 61] [4, 6, 10, 28, 52, 55, 62, 64] [5, 7, 11, 29, 53, 54, 61, 63] [6, 8, 12, 30, 53, 54, 64, 66] [7, 9, 13, 31, 52, 55, 63, 65] [8, 10, 14, 32, 51, 56, 66, 68] [9, 11, 15, 33, 50, 57, 65, 67] [0, 10, 12, 16, 34, 49, 68, 70] [1, 11, 13, 17, 35, 48, 67, 69] [2, 12, 14, 18, 36, 47, 70, 72] [3, 13, 15, 19, 37, 46, 69, 71] [4, 14, 16, 20, 38, 45, 72, 74] [5, 15, 17, 21, 39, 44, 71, 73] [6, 16, 18, 22, 40, 43, 74, 76] [7, 17, 19, 23, 41, 42, 73, 75] [8, 18, 20, 24, 41, 42, 76, 78] [9, 19, 21, 25, 40, 43, 75, 77] [10, 20, 22, 26, 39, 44, 78, 80] [11, 21, 23, 27, 38, 45, 77, 79] [12, 22, 24, 28, 37, 46, 80, 82] [13, 23, 25, 29, 36, 47, 79, 81] [14, 24, 26, 30, 35, 48, 82, 84] [15, 25, 27, 31, 34, 49, 81, 83] [16, 26, 28, 32, 33, 50, 84, 86] [17, 27, 29, 32, 33, 51, 83, 85] [18, 28, 30, 31, 34, 52, 86, 88] [19, 29, 30, 31, 35, 53, 85, 87] [20, 29, 30, 32, 36, 54, 88, 90] [21, 28, 31, 33, 37, 55, 87, 89] [22, 27, 32, 34, 38, 56, 90, 92] [23, 26, 33, 35, 39, 57, 89, 91] [0, 24, 25, 34, 36, 40, 92, 94] [1, 24, 25, 35, 37, 41, 91, 93] [2, 23, 26, 36, 38, 42, 94, 96] [3, 22, 27, 37, 39, 43, 93, 95] [4, 21, 28, 38, 40, 44, 96, 98] [5, 20, 29, 39, 41, 45, 95, 97] [6, 19, 30, 40, 42, 46, 98, 100] [7, 18, 31, 41, 43, 47, 97, 99] [8, 17, 32, 42, 44, 48, 100, 102] [9, 16, 33, 43, 45, 49, 99, 101] [10, 15, 34, 44, 46, 50, 102, 104] [11, 14, 35, 45, 47, 51, 101, 103] [12, 13, 36, 46, 48, 52, 104, 106] [12, 13, 37, 47, 49, 53, 103, 105] [11, 14, 38, 48, 50, 54, 106, 108] [10, 15, 39, 49, 51, 55, 105, 107] [9, 16, 40, 50, 52, 56, 108, 110] [8, 17, 41, 51, 53, 57, 107, 109] [0, 7, 18, 42, 52, 54, 110, 112] [1, 6, 19, 43, 53, 55, 109, 111] [2, 5, 20, 44, 54, 56, 112, 114] [3, 4, 21, 45, 55, 57, 111, 113] [0, 3, 4, 22, 46, 56, 58, 114] [1, 2, 5, 23, 47, 57, 113, 115]
Code ID 116-2-15 · download JSON · raw on GitHub