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[[116,4,15]] d ≤
n
116
k
4
d
15
kd²/n
7.759
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)
[0, 1, 10, 21, 47, 56, 59, 65, 69, 77, 80, 98, 100, 104, 105]
d_Z 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[5, 20, 27, 32, 42, 49, 54, 62, 70, 76, 84, 92, 98, 106, 114]
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): 348 (2,10): 174 (3,2): 58 (3,4): 464 (3,6): 3306 (3,8): 5800 (3,10): 6902 (3,12): 3712 (3,14): 870
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): 348 (2,10): 174 (3,2): 58 (3,4): 464 (3,6): 3306 (3,8): 5800 (3,10): 6902 (3,12): 3712 (3,14): 870

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_k4.txt, line 12. nonidentity supports a=[3], b=[2, 3, 10, 15, 47].
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

[[116,4,15]] — 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 = 7.76.

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_k4.txt, line 12. This row is SmallGroup(58,1) with nonidentity GAP supports a=[3], b=[2, 3, 10, 15, 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=4), 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, 10, 15, 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 = 116 - 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 58 (max weight 8) · Z-checks 58 (max weight 8)
H_X (58 checks, sparse supports)
[0, 56, 58, 59, 67, 70, 102, 114] [1, 3, 58, 59, 66, 71, 103, 115] [0, 2, 58, 60, 65, 72, 104, 115] [3, 5, 59, 61, 64, 73, 105, 114] [2, 4, 60, 62, 63, 74, 106, 113] [5, 7, 61, 62, 63, 75, 107, 112] [4, 6, 61, 62, 64, 76, 108, 111] [7, 9, 60, 63, 65, 77, 109, 110] [6, 8, 59, 64, 66, 78, 109, 110] [9, 11, 58, 65, 67, 79, 108, 111] [8, 10, 66, 68, 80, 107, 112, 115] [11, 13, 67, 69, 81, 106, 113, 114] [10, 12, 68, 70, 82, 105, 113, 114] [13, 15, 69, 71, 83, 104, 112, 115] [12, 14, 58, 70, 72, 84, 103, 111] [15, 17, 59, 71, 73, 85, 102, 110] [14, 16, 60, 72, 74, 86, 101, 109] [17, 19, 61, 73, 75, 87, 100, 108] [16, 18, 62, 74, 76, 88, 99, 107] [19, 21, 63, 75, 77, 89, 98, 106] [18, 20, 64, 76, 78, 90, 97, 105] [21, 23, 65, 77, 79, 91, 96, 104] [20, 22, 66, 78, 80, 92, 95, 103] [23, 25, 67, 79, 81, 93, 94, 102] [22, 24, 68, 80, 82, 93, 94, 101] [25, 27, 69, 81, 83, 92, 95, 100] [24, 26, 70, 82, 84, 91, 96, 99] [27, 29, 71, 83, 85, 90, 97, 98] [26, 28, 72, 84, 86, 89, 97, 98] [29, 31, 73, 85, 87, 88, 96, 99] [28, 30, 74, 86, 87, 88, 95, 100] [31, 33, 75, 86, 87, 89, 94, 101] [30, 32, 76, 85, 88, 90, 93, 102] [33, 35, 77, 84, 89, 91, 92, 103] [32, 34, 78, 83, 90, 91, 92, 104] [35, 37, 79, 82, 90, 91, 93, 105] [34, 36, 80, 81, 89, 92, 94, 106] [37, 39, 80, 81, 88, 93, 95, 107] [36, 38, 79, 82, 87, 94, 96, 108] [39, 41, 78, 83, 86, 95, 97, 109] [38, 40, 77, 84, 85, 96, 98, 110] [41, 43, 76, 84, 85, 97, 99, 111] [40, 42, 75, 83, 86, 98, 100, 112] [43, 45, 74, 82, 87, 99, 101, 113] [42, 44, 73, 81, 88, 100, 102, 114] [45, 47, 72, 80, 89, 101, 103, 115] [44, 46, 58, 71, 79, 90, 102, 104] [47, 49, 59, 70, 78, 91, 103, 105] [46, 48, 60, 69, 77, 92, 104, 106] [49, 51, 61, 68, 76, 93, 105, 107] [48, 50, 62, 67, 75, 94, 106, 108] [51, 53, 63, 66, 74, 95, 107, 109] [50, 52, 64, 65, 73, 96, 108, 110] [53, 55, 64, 65, 72, 97, 109, 111] [52, 54, 63, 66, 71, 98, 110, 112] [55, 57, 62, 67, 70, 99, 111, 113] [54, 56, 61, 68, 69, 100, 112, 114] [1, 57, 60, 68, 69, 101, 113, 115]
H_Z (58 checks, sparse supports)
[0, 1, 2, 9, 14, 46, 58, 60] [0, 1, 3, 8, 15, 47, 59, 115] [2, 4, 7, 16, 48, 57, 60, 62] [3, 5, 6, 17, 49, 56, 59, 61] [4, 5, 6, 18, 50, 55, 62, 64] [4, 5, 7, 19, 51, 54, 61, 63] [3, 6, 8, 20, 52, 53, 64, 66] [2, 7, 9, 21, 52, 53, 63, 65] [1, 8, 10, 22, 51, 54, 66, 68] [0, 9, 11, 23, 50, 55, 65, 67] [10, 12, 24, 49, 56, 57, 68, 70] [11, 13, 25, 48, 56, 57, 67, 69] [0, 12, 14, 26, 47, 55, 70, 72] [1, 13, 15, 27, 46, 54, 69, 71] [2, 14, 16, 28, 45, 53, 72, 74] [3, 15, 17, 29, 44, 52, 71, 73] [4, 16, 18, 30, 43, 51, 74, 76] [5, 17, 19, 31, 42, 50, 73, 75] [6, 18, 20, 32, 41, 49, 76, 78] [7, 19, 21, 33, 40, 48, 75, 77] [8, 20, 22, 34, 39, 47, 78, 80] [9, 21, 23, 35, 38, 46, 77, 79] [10, 22, 24, 36, 37, 45, 80, 82] [11, 23, 25, 36, 37, 44, 79, 81] [12, 24, 26, 35, 38, 43, 82, 84] [13, 25, 27, 34, 39, 42, 81, 83] [14, 26, 28, 33, 40, 41, 84, 86] [15, 27, 29, 32, 40, 41, 83, 85] [16, 28, 30, 31, 39, 42, 86, 88] [17, 29, 30, 31, 38, 43, 85, 87] [18, 29, 30, 32, 37, 44, 88, 90] [19, 28, 31, 33, 36, 45, 87, 89] [20, 27, 32, 34, 35, 46, 90, 92] [21, 26, 33, 34, 35, 47, 89, 91] [22, 25, 33, 34, 36, 48, 92, 94] [23, 24, 32, 35, 37, 49, 91, 93] [23, 24, 31, 36, 38, 50, 94, 96] [22, 25, 30, 37, 39, 51, 93, 95] [21, 26, 29, 38, 40, 52, 96, 98] [20, 27, 28, 39, 41, 53, 95, 97] [19, 27, 28, 40, 42, 54, 98, 100] [18, 26, 29, 41, 43, 55, 97, 99] [17, 25, 30, 42, 44, 56, 100, 102] [16, 24, 31, 43, 45, 57, 99, 101] [0, 15, 23, 32, 44, 46, 102, 104] [1, 14, 22, 33, 45, 47, 101, 103] [2, 13, 21, 34, 46, 48, 104, 106] [3, 12, 20, 35, 47, 49, 103, 105] [4, 11, 19, 36, 48, 50, 106, 108] [5, 10, 18, 37, 49, 51, 105, 107] [6, 9, 17, 38, 50, 52, 108, 110] [7, 8, 16, 39, 51, 53, 107, 109] [7, 8, 15, 40, 52, 54, 110, 112] [6, 9, 14, 41, 53, 55, 109, 111] [5, 10, 13, 42, 54, 56, 112, 114] [4, 11, 12, 43, 55, 57, 111, 113] [0, 3, 11, 12, 44, 56, 58, 114] [1, 2, 10, 13, 45, 57, 113, 115]
Code ID 116-4-15 · download JSON · raw on GitHub