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[[114,6,13]] d ≤
n
114
k
6
d
13
kd²/n
8.895
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 13, d_Z ≤ 13 · 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 13 · witness weight 13 (claimed upper_bound)
witness operator (support, 13 qubits)
[3, 9, 14, 16, 31, 47, 56, 62, 74, 84, 97, 104, 107]
d_Z 13 · witness weight 13 (claimed upper_bound)
witness operator (support, 13 qubits)
[1, 2, 3, 4, 18, 22, 25, 29, 32, 35, 62, 103, 109]
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 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)×57 (2,2)×57 (3,2)×57 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 57 (1,6): 57 (2,2): 57 (2,6): 684 (2,10): 855 (3,2): 57 (3,4): 19 (3,6): 4047 (3,8): 1292 (3,10): 16302 (3,12): 4845 (3,14): 11970 (3,16): 1140
trapping sets H_Z (1,2)×57 (2,2)×57 (3,2)×57 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 57 (1,6): 57 (2,2): 57 (2,6): 684 (2,10): 855 (3,2): 57 (3,4): 19 (3,6): 4047 (3,8): 1292 (3,10): 16302 (3,12): 4845 (3,14): 11970 (3,16): 1140

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(57,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order57_k6.txt, line 10. nonidentity supports a=[3], b=[2, 6, 20, 25, 52].
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

[[114,6,13]] — two-block group-algebra code on SmallGroup(57,1)

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 = 8.89.

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_order57_k6.txt, line 10. This row is SmallGroup(57,1) with nonidentity GAP supports a=[3], b=[2, 6, 20, 25, 52]. 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=6), 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 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(57,1), 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, 6, 20, 25, 52] 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 = 114 - rank(HX) - rank(HZ) = 6. 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 57 (max weight 8) · Z-checks 57 (max weight 8)
H_X (57 checks, sparse supports)
[0, 53, 57, 60, 64, 82, 90, 107] [1, 37, 57, 58, 66, 84, 86, 109] [0, 2, 59, 66, 85, 93, 103, 110] [3, 27, 58, 60, 62, 80, 88, 111] [4, 40, 61, 62, 87, 89, 105, 112] [2, 5, 57, 62, 67, 69, 99, 106] [6, 30, 63, 64, 83, 91, 101, 113] [7, 43, 58, 64, 65, 69, 95, 108] [5, 8, 59, 65, 70, 75, 88, 102] [9, 33, 60, 65, 66, 67, 97, 104] [10, 46, 61, 67, 71, 72, 90, 98] [8, 11, 62, 68, 78, 91, 108, 109] [12, 36, 63, 68, 69, 73, 86, 100] [13, 49, 64, 70, 74, 93, 104, 111] [11, 14, 65, 71, 73, 84, 111, 112] [15, 39, 66, 72, 76, 89, 106, 107] [16, 52, 67, 73, 75, 80, 107, 113] [14, 17, 60, 68, 74, 76, 87, 94] [18, 42, 69, 71, 75, 82, 109, 110] [19, 55, 57, 70, 76, 78, 83, 96] [17, 20, 58, 63, 71, 77, 93, 97] [21, 45, 58, 72, 74, 78, 85, 92] [1, 22, 59, 60, 73, 79, 89, 99] [20, 23, 61, 69, 74, 79, 80, 96] [24, 48, 57, 61, 75, 81, 91, 95] [4, 25, 63, 65, 76, 81, 82, 92] [23, 26, 72, 77, 82, 83, 100, 102] [27, 51, 59, 67, 77, 78, 84, 94] [7, 28, 68, 79, 84, 85, 98, 102] [26, 29, 64, 78, 80, 86, 103, 105] [30, 54, 70, 80, 81, 87, 98, 100] [10, 31, 66, 74, 82, 88, 101, 105] [29, 32, 67, 81, 83, 85, 89, 111] [33, 56, 62, 76, 84, 90, 101, 103] [13, 34, 69, 77, 85, 87, 91, 107] [32, 35, 86, 87, 88, 92, 106, 113] [3, 36, 65, 79, 83, 87, 93, 109] [16, 37, 83, 88, 90, 94, 108, 110] [35, 38, 63, 70, 89, 90, 95, 109] [6, 39, 85, 86, 90, 96, 104, 112] [19, 40, 59, 72, 86, 91, 97, 111] [38, 41, 66, 73, 91, 92, 96, 98] [9, 42, 61, 68, 88, 93, 99, 107] [22, 43, 62, 75, 92, 93, 94, 100] [41, 44, 72, 94, 95, 99, 101, 112] [12, 45, 64, 71, 89, 94, 96, 102] [25, 46, 68, 95, 96, 97, 103, 113] [44, 47, 58, 75, 76, 98, 104, 105] [15, 48, 70, 92, 97, 99, 105, 110] [28, 49, 60, 71, 78, 100, 101, 106] [47, 50, 79, 81, 97, 101, 107, 108] [18, 51, 57, 73, 74, 102, 103, 108] [31, 52, 77, 81, 99, 103, 104, 109] [50, 53, 61, 84, 100, 104, 110, 113] [21, 54, 77, 79, 95, 105, 106, 111] [34, 55, 63, 80, 102, 106, 110, 112] [24, 56, 59, 82, 98, 108, 112, 113]
H_Z (57 checks, sparse supports)
[0, 1, 5, 19, 24, 51, 57, 59] [1, 3, 7, 20, 21, 47, 58, 79] [2, 8, 22, 27, 40, 56, 59, 62] [0, 3, 9, 17, 22, 49, 60, 93] [4, 10, 23, 24, 42, 53, 61, 82] [3, 4, 5, 11, 33, 43, 62, 65] [6, 12, 20, 25, 38, 55, 63, 96] [0, 6, 7, 13, 29, 45, 64, 85] [7, 8, 9, 14, 25, 36, 65, 68] [1, 2, 9, 15, 31, 41, 66, 99] [5, 9, 10, 16, 27, 32, 67, 88] [11, 12, 17, 28, 42, 46, 68, 71] [5, 7, 12, 18, 23, 34, 69, 102] [8, 13, 19, 30, 38, 48, 70, 91] [10, 14, 18, 20, 45, 49, 71, 74] [10, 15, 21, 26, 40, 44, 72, 105] [12, 14, 16, 22, 41, 51, 73, 94] [13, 17, 21, 23, 31, 51, 74, 77] [8, 16, 18, 24, 43, 47, 75, 108] [15, 17, 19, 25, 33, 47, 76, 97] [20, 26, 27, 34, 52, 54, 77, 80] [11, 19, 21, 27, 29, 49, 78, 111] [22, 23, 28, 36, 50, 54, 79, 100] [3, 16, 23, 29, 30, 55, 80, 83] [24, 25, 30, 32, 50, 52, 81, 113] [0, 18, 25, 26, 31, 56, 82, 103] [6, 19, 26, 32, 36, 37, 83, 86] [1, 14, 27, 28, 33, 53, 60, 84] [2, 21, 28, 32, 34, 39, 85, 106] [1, 12, 29, 35, 39, 40, 86, 89] [4, 17, 30, 34, 35, 36, 63, 87] [3, 8, 31, 35, 37, 42, 88, 109] [4, 15, 22, 32, 38, 45, 89, 92] [0, 10, 33, 37, 38, 39, 66, 90] [6, 11, 24, 34, 40, 41, 91, 112] [21, 25, 35, 41, 43, 48, 92, 95] [2, 13, 20, 36, 42, 43, 69, 93] [17, 27, 37, 43, 44, 45, 58, 94] [7, 24, 38, 44, 46, 54, 95, 98] [19, 23, 39, 41, 45, 46, 72, 96] [9, 20, 40, 46, 48, 50, 61, 97] [10, 28, 30, 41, 47, 56, 98, 101] [5, 22, 42, 44, 48, 52, 75, 99] [12, 26, 30, 43, 49, 53, 64, 100] [6, 31, 33, 44, 49, 50, 101, 104] [8, 26, 28, 45, 51, 55, 78, 102] [2, 29, 33, 46, 51, 52, 67, 103] [9, 13, 39, 47, 52, 53, 104, 107] [4, 29, 31, 47, 48, 54, 81, 105] [5, 15, 35, 49, 54, 55, 70, 106] [0, 15, 16, 34, 42, 50, 107, 110] [7, 11, 37, 50, 51, 56, 84, 108] [1, 11, 18, 36, 38, 52, 73, 109] [2, 18, 37, 48, 53, 55, 57, 110] [3, 13, 14, 32, 40, 54, 87, 111] [4, 14, 39, 44, 55, 56, 76, 112] [6, 16, 35, 46, 53, 56, 90, 113]
Code ID 114-6-13 · download JSON · raw on GitHub