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[[112,8,13]] d ≤
n
112
k
8
d
13
kd²/n
12.071
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)
[4, 16, 21, 37, 38, 48, 50, 60, 66, 68, 73, 77, 85]
d_Z 13 · witness weight 13 (claimed upper_bound)
witness operator (support, 13 qubits)
[11, 17, 23, 24, 31, 37, 65, 71, 74, 88, 96, 97, 99]
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 4 · H_Z 4
trapping sets H_X (1,4)×112 (2,4)×112 (3,4)×224 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 112 (2,4): 112 (2,6): 1344 (3,4): 224 (3,6): 3640 (3,8): 21728 (3,10): 1904
trapping sets H_Z (1,4)×112 (2,4)×112 (3,4)×224 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 112 (2,4): 112 (2,6): 1344 (3,4): 224 (3,6): 3640 (3,8): 21728 (3,10): 1904

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(56,3), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order56_k8.txt, line 10. nonidentity supports a=[2, 10, 22], b=[3, 17, 19].
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

[[112,8,13]] — two-block group-algebra code on SmallGroup(56,3)

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 = 12.07.

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_wt8wtL4_order56_k8.txt, line 10. This row is SmallGroup(56,3) with nonidentity GAP supports a=[2, 10, 22], b=[3, 17, 19]. 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 ≤ 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(56,3), 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=[2, 10, 22], b=[3, 17, 19] 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) = 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 56 (max weight 8) · Z-checks 56 (max weight 8)
H_X (56 checks, sparse supports)
[0, 6, 28, 54, 56, 64, 98, 105] [0, 1, 13, 48, 57, 68, 102, 108] [2, 12, 15, 43, 56, 58, 104, 106] [1, 3, 21, 49, 58, 59, 91, 110] [4, 8, 14, 36, 58, 60, 72, 106] [2, 5, 14, 35, 57, 61, 107, 109] [3, 6, 20, 41, 61, 62, 95, 111] [4, 7, 21, 54, 61, 63, 76, 109] [5, 8, 22, 50, 59, 64, 97, 99] [0, 9, 20, 23, 59, 60, 65, 110] [2, 7, 10, 29, 64, 65, 66, 99] [11, 16, 22, 44, 59, 65, 67, 80] [7, 8, 12, 42, 62, 68, 101, 103] [9, 13, 22, 43, 62, 63, 69, 111] [10, 14, 28, 49, 68, 69, 70, 103] [8, 11, 15, 29, 62, 69, 71, 84] [3, 13, 16, 30, 56, 66, 72, 105] [4, 17, 28, 31, 64, 66, 67, 73] [9, 15, 18, 37, 56, 72, 73, 74] [19, 24, 30, 51, 66, 73, 75, 88] [15, 16, 20, 50, 57, 70, 76, 108] [0, 17, 21, 30, 68, 70, 71, 77] [2, 18, 22, 36, 57, 76, 77, 78] [16, 19, 23, 37, 70, 77, 79, 92] [10, 21, 24, 38, 58, 60, 74, 80] [11, 25, 36, 39, 72, 74, 75, 81] [17, 23, 26, 45, 60, 80, 81, 82] [27, 32, 38, 55, 74, 81, 83, 96] [3, 23, 24, 28, 61, 63, 78, 84] [4, 25, 29, 38, 76, 78, 79, 85] [9, 26, 30, 44, 63, 84, 85, 86] [24, 27, 31, 45, 78, 85, 87, 100] [18, 29, 32, 46, 65, 67, 82, 88] [19, 33, 44, 47, 80, 82, 83, 89] [25, 31, 34, 52, 67, 88, 89, 90] [12, 35, 40, 46, 82, 89, 91, 104] [10, 31, 32, 36, 69, 71, 86, 92] [11, 33, 37, 46, 84, 86, 87, 93] [17, 34, 38, 51, 71, 92, 93, 94] [32, 35, 39, 52, 86, 93, 95, 107] [26, 37, 40, 53, 73, 75, 90, 96] [1, 27, 41, 51, 88, 90, 91, 97] [5, 33, 39, 42, 75, 96, 97, 98] [20, 43, 48, 53, 90, 97, 99, 110] [18, 39, 40, 44, 77, 79, 94, 100] [19, 41, 45, 53, 92, 94, 95, 101] [25, 42, 46, 55, 79, 100, 101, 102] [5, 40, 43, 47, 94, 101, 103, 111] [6, 34, 45, 48, 81, 83, 98, 104] [7, 35, 49, 55, 96, 98, 99, 105] [13, 41, 47, 50, 83, 104, 105, 106] [26, 47, 48, 51, 85, 87, 102, 107] [6, 27, 49, 52, 100, 102, 103, 108] [12, 33, 50, 53, 87, 107, 108, 109] [14, 42, 52, 54, 89, 91, 106, 110] [1, 34, 54, 55, 93, 95, 109, 111]
H_Z (56 checks, sparse supports)
[0, 2, 16, 18, 56, 57, 65, 77] [1, 5, 20, 22, 57, 59, 97, 111] [2, 3, 4, 24, 58, 61, 66, 78] [3, 8, 9, 11, 59, 62, 72, 84] [4, 9, 24, 26, 60, 63, 73, 85] [5, 6, 7, 28, 61, 64, 98, 103] [6, 12, 13, 15, 56, 62, 104, 108] [7, 13, 28, 30, 63, 66, 68, 105] [0, 8, 10, 17, 60, 64, 68, 71] [9, 10, 11, 32, 65, 69, 74, 86] [10, 16, 17, 19, 66, 70, 80, 92] [11, 17, 32, 34, 67, 71, 81, 93] [1, 12, 14, 21, 58, 68, 91, 109] [13, 14, 15, 36, 57, 69, 72, 106] [14, 20, 21, 23, 60, 61, 70, 110] [15, 21, 36, 38, 58, 71, 74, 76] [4, 16, 18, 25, 67, 72, 76, 79] [17, 18, 19, 40, 73, 77, 82, 94] [18, 24, 25, 27, 74, 78, 88, 100] [19, 25, 40, 42, 75, 79, 89, 101] [7, 20, 22, 29, 62, 65, 76, 99] [21, 22, 23, 44, 59, 63, 77, 80] [22, 28, 29, 31, 64, 67, 69, 78] [23, 29, 44, 46, 65, 79, 82, 84] [11, 24, 26, 33, 75, 80, 84, 87] [25, 26, 27, 48, 81, 85, 90, 102] [26, 32, 33, 35, 82, 86, 96, 107] [27, 33, 48, 50, 83, 87, 97, 108] [15, 28, 30, 37, 56, 70, 73, 84] [29, 30, 31, 51, 66, 71, 85, 88] [30, 36, 37, 39, 72, 75, 77, 86] [31, 37, 51, 53, 73, 87, 90, 92] [19, 32, 34, 41, 83, 88, 92, 95] [33, 34, 35, 54, 89, 93, 98, 109] [34, 40, 41, 43, 90, 94, 104, 111] [3, 35, 41, 54, 61, 91, 95, 105] [23, 36, 38, 45, 60, 78, 81, 92] [37, 38, 39, 55, 74, 79, 93, 96] [38, 44, 45, 47, 80, 83, 85, 94] [6, 39, 45, 55, 81, 95, 98, 100] [27, 40, 42, 49, 91, 96, 100, 103] [8, 41, 42, 43, 62, 97, 101, 106] [0, 42, 48, 49, 68, 98, 102, 110] [8, 10, 43, 49, 58, 69, 99, 103] [31, 44, 46, 52, 67, 86, 89, 100] [12, 45, 46, 47, 82, 87, 101, 104] [1, 46, 51, 52, 88, 91, 93, 102] [12, 14, 47, 52, 89, 103, 106, 107] [2, 35, 48, 50, 57, 99, 104, 107] [0, 16, 49, 50, 59, 70, 105, 108] [2, 4, 50, 54, 64, 76, 106, 109] [5, 39, 51, 53, 75, 94, 97, 107] [1, 20, 52, 53, 90, 95, 108, 110] [5, 7, 53, 55, 96, 99, 101, 109] [3, 9, 43, 54, 56, 63, 110, 111] [6, 13, 47, 55, 83, 102, 105, 111]
Code ID 112-8-13 · download JSON · raw on GitHub