← back to the board
[[112,2,16]] d ≤
n
112
k
2
d
16
kd²/n
4.571
w
8
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[10, 24, 32, 35, 38, 48, 64, 66, 74, 79, 84, 86, 91, 94, 97, 98]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[3, 6, 12, 19, 30, 31, 43, 44, 45, 51, 57, 61, 68, 76, 94, 110]
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)×56 (2,2)×56 (3,2)×56 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 56 (1,6): 56 (2,2): 56 (2,6): 672 (2,8): 168 (2,10): 504 (3,2): 56 (3,4): 56 (3,6): 3920 (3,8): 3416 (3,10): 12488 (3,12): 4872 (3,14): 5488 (3,16): 224
trapping sets H_Z (1,2)×56 (2,2)×56 (3,2)×56 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 56 (1,6): 56 (2,2): 56 (2,6): 672 (2,8): 168 (2,10): 504 (3,2): 56 (3,4): 56 (3,6): 3920 (3,8): 3416 (3,10): 12488 (3,12): 4872 (3,14): 5488 (3,16): 224

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_wt8wtL2_order56_k2.txt, line 10. nonidentity supports a=[10], b=[2, 3, 5, 12, 43].
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

[[112,2,16]] — 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 = 4.57.

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_order56_k2.txt, line 10. This row is SmallGroup(56,3) with nonidentity GAP supports a=[10], b=[2, 3, 5, 12, 43]. 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 ≤ 16, 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=[10], b=[2, 3, 5, 12, 43] 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) = 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 56 (max weight 8) · Z-checks 56 (max weight 8)
H_X (56 checks, sparse supports)
[0, 54, 56, 62, 64, 74, 91, 99] [1, 13, 56, 57, 68, 78, 95, 103] [2, 43, 56, 58, 61, 80, 97, 105] [3, 49, 57, 58, 59, 67, 98, 106] [4, 8, 56, 60, 72, 82, 99, 109] [5, 14, 57, 61, 64, 84, 101, 108] [6, 20, 59, 61, 62, 71, 102, 109] [7, 21, 57, 63, 76, 86, 99, 103] [8, 50, 59, 64, 68, 73, 104, 110] [0, 9, 58, 60, 65, 88, 105, 108] [2, 10, 59, 65, 66, 75, 103, 106] [11, 16, 56, 60, 67, 80, 90, 102] [7, 12, 58, 62, 68, 77, 107, 111] [13, 22, 61, 63, 69, 92, 108, 110] [14, 28, 62, 69, 70, 79, 106, 109] [15, 29, 57, 63, 71, 84, 91, 94] [3, 16, 64, 66, 72, 81, 110, 111] [4, 17, 58, 65, 67, 73, 96, 101] [9, 18, 59, 66, 73, 74, 83, 95] [19, 24, 60, 67, 75, 88, 94, 98] [15, 20, 68, 70, 76, 85, 105, 111] [21, 30, 61, 69, 71, 77, 100, 104] [22, 36, 62, 70, 77, 78, 87, 98] [23, 37, 63, 71, 79, 83, 92, 102] [10, 24, 64, 72, 74, 80, 89, 107] [11, 25, 65, 73, 75, 81, 93, 104] [17, 26, 66, 74, 81, 82, 87, 91] [27, 32, 67, 75, 83, 86, 96, 106] [23, 28, 68, 76, 78, 84, 93, 97] [29, 38, 69, 77, 79, 85, 96, 107] [30, 44, 70, 78, 85, 86, 90, 95] [31, 45, 71, 75, 79, 87, 100, 109] [18, 32, 72, 80, 82, 88, 97, 100] [19, 33, 73, 81, 83, 85, 89, 110] [25, 34, 74, 79, 82, 89, 90, 99] [35, 40, 59, 75, 78, 83, 91, 104] [31, 36, 76, 84, 86, 89, 92, 101] [37, 46, 77, 85, 87, 88, 93, 111] [38, 51, 78, 82, 86, 93, 94, 103] [39, 52, 62, 67, 79, 87, 95, 107] [26, 40, 80, 88, 90, 92, 96, 105] [27, 41, 64, 77, 81, 89, 91, 97] [33, 42, 56, 71, 82, 90, 97, 98] [43, 48, 66, 70, 83, 91, 99, 110] [39, 44, 81, 84, 92, 94, 100, 108] [45, 53, 68, 80, 85, 93, 95, 101] [46, 55, 57, 74, 86, 94, 101, 102] [5, 47, 60, 70, 87, 95, 103, 111] [34, 48, 58, 84, 88, 96, 98, 104] [35, 49, 69, 72, 89, 97, 99, 105] [41, 50, 60, 63, 90, 98, 105, 106] [47, 51, 61, 73, 92, 100, 102, 107] [6, 52, 72, 76, 93, 101, 103, 108] [12, 53, 63, 66, 94, 102, 108, 109] [42, 54, 65, 76, 96, 104, 106, 110] [1, 55, 65, 69, 100, 107, 109, 111]
H_Z (56 checks, sparse supports)
[0, 1, 2, 4, 11, 42, 56, 65] [1, 3, 5, 7, 15, 46, 57, 111] [2, 3, 9, 12, 17, 48, 58, 66] [3, 6, 8, 10, 18, 35, 59, 72] [4, 9, 11, 19, 47, 50, 60, 73] [2, 5, 6, 13, 21, 51, 61, 103] [0, 6, 12, 14, 22, 39, 62, 108] [7, 13, 15, 23, 50, 53, 63, 68] [0, 5, 8, 16, 24, 41, 60, 64] [9, 10, 17, 25, 54, 55, 65, 74] [10, 16, 18, 26, 43, 53, 66, 80] [3, 11, 17, 19, 27, 39, 67, 81] [1, 8, 12, 20, 28, 45, 68, 109] [13, 14, 21, 29, 49, 55, 57, 69] [14, 20, 22, 30, 43, 47, 61, 70] [6, 15, 21, 23, 31, 42, 71, 76] [4, 16, 24, 32, 49, 52, 67, 72] [8, 17, 18, 25, 33, 51, 73, 82] [0, 18, 24, 26, 34, 46, 74, 88] [10, 19, 25, 27, 31, 35, 75, 89] [7, 20, 28, 36, 52, 54, 62, 76] [12, 21, 22, 29, 37, 41, 63, 77] [1, 22, 28, 30, 35, 38, 69, 78] [14, 23, 29, 31, 34, 39, 79, 84] [2, 11, 24, 32, 40, 45, 75, 80] [16, 25, 26, 33, 41, 44, 81, 90] [4, 26, 32, 34, 38, 42, 82, 96] [18, 23, 27, 33, 35, 43, 83, 97] [5, 15, 28, 36, 44, 48, 70, 84] [20, 29, 30, 33, 37, 45, 71, 85] [7, 27, 30, 36, 38, 46, 77, 86] [22, 26, 31, 37, 39, 47, 87, 92] [9, 19, 32, 37, 40, 48, 83, 88] [24, 33, 34, 36, 41, 49, 89, 98] [11, 30, 34, 40, 42, 50, 90, 104] [0, 15, 26, 35, 41, 43, 91, 105] [13, 23, 36, 40, 44, 51, 78, 92] [25, 28, 37, 38, 45, 52, 79, 93] [15, 19, 38, 44, 46, 53, 85, 94] [1, 18, 30, 39, 45, 47, 95, 100] [17, 27, 29, 40, 48, 54, 91, 96] [2, 28, 32, 41, 42, 49, 97, 106] [3, 19, 22, 42, 48, 50, 98, 110] [0, 4, 7, 34, 43, 49, 58, 99] [21, 31, 32, 44, 51, 55, 86, 100] [5, 17, 36, 45, 46, 52, 87, 101] [6, 11, 23, 46, 51, 53, 93, 102] [1, 7, 10, 38, 47, 52, 103, 107] [8, 21, 25, 35, 48, 54, 99, 104] [2, 9, 20, 40, 49, 50, 59, 105] [3, 10, 14, 27, 50, 54, 64, 106] [12, 24, 29, 39, 51, 55, 94, 107] [5, 9, 13, 44, 52, 53, 95, 108] [4, 6, 14, 31, 53, 55, 101, 109] [8, 13, 16, 33, 43, 54, 56, 110] [12, 16, 20, 37, 47, 55, 102, 111]
Code ID 112-2-16 · download JSON · raw on GitHub