← back to the board
[[120,8,14]] d ≤
n
120
k
8
d
14
kd²/n
13.067
w
7
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[2, 16, 35, 58, 62, 69, 71, 82, 83, 90, 91, 95, 103, 106]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[3, 8, 27, 31, 43, 44, 54, 59, 75, 83, 88, 94, 95, 118]
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 7 · H_Z 7
qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)
trapping sets H_X (1,3)×60 (2,4)×180 (3,3)×60 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 60 (1,4): 60 (2,4): 180 (2,5): 720 (2,6): 360 (3,3): 60 (3,4): 180 (3,5): 780 (3,6): 5880 (3,7): 8700 (3,8): 3060 (3,9): 1080 (3,10): 240
trapping sets H_Z (1,3)×60 (2,4)×180 (3,3)×60 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 60 (1,4): 60 (2,4): 180 (2,5): 720 (2,6): 360 (3,3): 60 (3,4): 180 (3,5): 780 (3,6): 5880 (3,7): 8700 (3,8): 3060 (3,9): 1080 (3,10): 240

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction contributed via qldpc submit
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-18
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

[[120,8,14]] — two-block group-algebra code on SmallGroup(60,2)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 13.07.

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_wt7wtL3_order60_k8.txt, line 6. This row is SmallGroup(60,2) with nonidentity GAP supports a=[5, 45], b=[2, 10, 20]. 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 ≤ 14, not an exact certificate. The published distance (d=14) 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

Union Alpha (maker currently anonymous), 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(60,2), 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=[5, 45] and b=[2, 10, 20] 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 = 120 - 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 60 (max weight 7) · Z-checks 60 (max weight 7)
H_X (60 checks, sparse supports)
[0, 35, 42, 60, 66, 78, 105] [1, 7, 55, 60, 61, 84, 110] [2, 22, 45, 62, 63, 74, 113] [3, 31, 46, 61, 63, 68, 114] [0, 4, 52, 62, 64, 90, 111] [5, 15, 40, 62, 65, 66, 116] [6, 16, 48, 63, 66, 73, 117] [7, 17, 59, 65, 67, 95, 96] [8, 32, 53, 68, 69, 84, 95] [9, 12, 54, 60, 65, 69, 118] [2, 10, 34, 68, 70, 71, 117] [3, 11, 43, 69, 71, 79, 101] [4, 12, 58, 70, 72, 100, 102] [13, 25, 49, 68, 73, 74, 101] [14, 26, 29, 61, 69, 74, 119] [15, 27, 51, 73, 75, 76, 105] [16, 28, 56, 74, 76, 85, 106] [17, 24, 29, 75, 77, 83, 107] [18, 21, 58, 62, 73, 78, 106] [8, 19, 44, 60, 79, 80, 119] [9, 20, 23, 64, 78, 80, 110] [10, 21, 46, 79, 81, 82, 109] [11, 22, 53, 80, 82, 89, 91] [12, 18, 23, 81, 83, 88, 112] [24, 36, 39, 65, 78, 84, 111] [25, 37, 57, 61, 85, 86, 113] [26, 38, 41, 67, 84, 86, 114] [6, 27, 39, 85, 87, 88, 93] [13, 28, 40, 86, 88, 94, 97] [29, 36, 41, 72, 87, 89, 115] [18, 30, 33, 63, 70, 90, 116] [19, 31, 54, 64, 91, 92, 115] [20, 32, 35, 72, 90, 92, 99] [3, 21, 33, 91, 93, 94, 98] [8, 22, 34, 77, 92, 94, 103] [23, 30, 35, 76, 93, 95, 118] [36, 47, 50, 66, 75, 96, 118] [14, 37, 48, 67, 97, 98, 103] [1, 38, 49, 77, 96, 98, 104] [16, 39, 50, 81, 97, 99, 100] [25, 40, 51, 82, 98, 100, 108] [1, 41, 47, 64, 99, 101, 119] [30, 42, 45, 71, 81, 102, 108] [9, 31, 43, 72, 103, 104, 107] [0, 32, 44, 83, 87, 102, 104] [11, 33, 45, 86, 103, 105, 106] [19, 34, 46, 67, 104, 106, 113] [5, 47, 55, 76, 87, 107, 112] [26, 48, 56, 77, 91, 108, 109] [7, 49, 57, 89, 92, 107, 109] [5, 28, 50, 70, 108, 110, 111] [6, 37, 51, 71, 109, 111, 116] [2, 42, 52, 82, 93, 97, 112] [20, 43, 53, 83, 96, 113, 114] [4, 44, 54, 75, 95, 112, 114] [15, 55, 59, 88, 99, 102, 115] [13, 38, 56, 79, 89, 116, 117] [14, 17, 57, 80, 101, 115, 117] [10, 52, 58, 85, 94, 105, 118] [24, 27, 59, 90, 100, 110, 119]
H_Z (60 checks, sparse supports)
[0, 1, 9, 19, 60, 64, 104] [1, 3, 14, 25, 61, 98, 101] [2, 4, 5, 18, 62, 70, 112] [2, 3, 6, 30, 63, 71, 93] [4, 20, 31, 41, 64, 72, 114] [5, 7, 9, 24, 65, 107, 110] [0, 5, 6, 36, 66, 87, 111] [7, 26, 37, 46, 61, 67, 109] [3, 8, 10, 13, 68, 79, 94] [8, 9, 11, 14, 69, 80, 103] [10, 12, 30, 50, 70, 81, 118] [10, 11, 42, 51, 71, 82, 105] [12, 29, 32, 43, 69, 72, 83] [6, 13, 15, 18, 73, 88, 116] [2, 13, 14, 16, 74, 97, 117] [15, 17, 36, 54, 65, 75, 115] [15, 16, 35, 47, 66, 76, 99] [17, 34, 38, 48, 67, 77, 117] [0, 18, 20, 24, 78, 83, 90] [11, 19, 21, 56, 79, 91, 106] [19, 20, 22, 57, 80, 92, 113] [21, 23, 39, 42, 78, 81, 93] [21, 22, 40, 52, 62, 82, 94] [17, 23, 44, 53, 80, 83, 95] [1, 8, 24, 26, 77, 84, 119] [16, 25, 27, 58, 73, 85, 100] [25, 26, 28, 45, 74, 86, 108] [27, 29, 44, 47, 75, 87, 119] [23, 27, 28, 55, 76, 88, 110] [22, 29, 49, 56, 74, 77, 89] [4, 30, 32, 59, 90, 95, 102] [22, 31, 33, 48, 63, 91, 103] [31, 32, 34, 49, 68, 92, 104] [27, 33, 35, 52, 90, 93, 105] [28, 33, 34, 58, 70, 94, 106] [7, 8, 35, 54, 60, 92, 95] [7, 36, 38, 53, 84, 89, 96] [28, 37, 39, 52, 85, 97, 111] [33, 37, 38, 40, 86, 98, 116] [32, 39, 41, 55, 84, 87, 99] [12, 39, 40, 59, 65, 88, 100] [11, 13, 41, 57, 86, 89, 101] [12, 42, 44, 55, 60, 102, 112] [34, 37, 43, 45, 71, 103, 113] [38, 43, 44, 46, 79, 104, 114] [0, 15, 45, 58, 62, 102, 105] [16, 18, 45, 46, 63, 81, 106] [17, 43, 47, 49, 96, 101, 107] [40, 42, 48, 50, 66, 97, 108] [21, 48, 49, 51, 73, 98, 109] [1, 20, 50, 59, 96, 99, 110] [4, 24, 50, 51, 75, 100, 111] [23, 47, 52, 54, 64, 112, 118] [2, 25, 46, 53, 68, 82, 113] [3, 26, 53, 54, 69, 91, 114] [29, 31, 55, 57, 61, 107, 115] [5, 30, 51, 56, 76, 108, 116] [6, 10, 56, 57, 85, 109, 117] [9, 35, 36, 58, 72, 78, 118] [14, 19, 41, 59, 67, 115, 119]
Code ID 120-8-14 · download JSON · raw on GitHub