← back to the board
[[120,6,15]] d ≤
n
120
k
6
d
15
kd²/n
11.25
w
8
X/Z
1

Share this result

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)
[20, 23, 36, 40, 63, 64, 74, 83, 84, 85, 92, 95, 96, 107, 116]
d_Z 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[8, 31, 35, 41, 42, 47, 48, 57, 58, 74, 91, 93, 107, 108, 111]
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)×60 (2,2)×60 (3,2)×60 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 60 (1,6): 60 (2,2): 60 (2,6): 720 (2,10): 900 (3,2): 60 (3,4): 60 (3,6): 4140 (3,8): 1260 (3,10): 17460 (3,12): 5180 (3,14): 12360 (3,16): 1200
trapping sets H_Z (1,2)×60 (2,2)×60 (3,2)×60 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 60 (1,6): 60 (2,2): 60 (2,6): 720 (2,10): 900 (3,2): 60 (3,4): 60 (3,6): 4140 (3,8): 1260 (3,10): 17460 (3,12): 5180 (3,14): 12360 (3,16): 1200

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(60,3), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order60_k6.txt, line 40. nonidentity supports a=[12], b=[2, 4, 10, 26, 46].
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

[[120,6,15]] — two-block group-algebra code on SmallGroup(60,3)

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 = 11.25.

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_order60_k6.txt, line 40. This row is SmallGroup(60,3) with nonidentity GAP supports a=[12], b=[2, 4, 10, 26, 46]. 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 ≤ 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(60,3), 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=[12], b=[2, 4, 10, 26, 46] 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) = 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 60 (max weight 8) · Z-checks 60 (max weight 8)
H_X (60 checks, sparse supports)
[0, 54, 60, 65, 70, 76, 82, 104] [1, 16, 60, 61, 75, 79, 88, 109] [2, 58, 61, 62, 78, 85, 91, 95] [3, 35, 60, 63, 67, 84, 93, 113] [4, 10, 62, 64, 66, 81, 94, 109] [5, 25, 62, 65, 71, 84, 97, 101] [6, 27, 61, 66, 69, 70, 99, 116] [7, 28, 65, 67, 68, 87, 95, 100] [8, 44, 62, 68, 74, 75, 102, 106] [9, 18, 60, 69, 73, 90, 101, 103] [10, 46, 63, 70, 72, 73, 87, 118] [0, 11, 61, 64, 68, 71, 105, 119] [12, 21, 69, 72, 93, 98, 106, 111] [13, 36, 64, 65, 73, 78, 107, 111] [14, 37, 61, 63, 74, 96, 104, 108] [7, 15, 63, 66, 75, 77, 90, 119] [16, 39, 62, 67, 73, 76, 110, 114] [17, 40, 74, 77, 83, 99, 111, 113] [18, 53, 66, 68, 78, 80, 96, 114] [2, 19, 63, 65, 69, 79, 112, 117] [20, 30, 64, 80, 89, 102, 113, 116] [3, 21, 71, 75, 78, 81, 83, 116] [4, 22, 72, 79, 82, 101, 114, 115] [23, 33, 63, 80, 83, 86, 100, 105] [14, 24, 68, 73, 81, 84, 86, 117] [25, 47, 60, 66, 74, 85, 115, 118] [26, 48, 67, 86, 92, 106, 107, 116] [17, 27, 76, 78, 84, 87, 89, 106] [28, 50, 77, 85, 88, 104, 105, 117] [29, 51, 66, 72, 86, 89, 103, 110] [8, 30, 70, 79, 84, 90, 92, 111] [9, 31, 71, 80, 91, 109, 110, 118] [32, 42, 68, 72, 77, 92, 108, 112] [11, 33, 82, 90, 93, 95, 108, 117] [12, 34, 70, 83, 89, 91, 94, 107] [35, 45, 71, 74, 88, 92, 95, 114] [26, 36, 70, 75, 85, 96, 98, 113] [37, 55, 76, 86, 95, 97, 112, 119] [38, 56, 73, 77, 80, 94, 98, 115] [29, 39, 88, 94, 96, 99, 101, 118] [40, 57, 75, 89, 92, 93, 97, 100] [6, 41, 64, 76, 91, 98, 101, 117] [19, 42, 81, 91, 100, 102, 104, 119] [20, 43, 78, 82, 92, 98, 99, 103] [44, 52, 67, 79, 83, 97, 104, 118] [22, 45, 60, 94, 97, 102, 105, 110] [23, 46, 77, 81, 95, 96, 103, 106] [38, 47, 87, 97, 103, 107, 109, 114] [48, 59, 83, 84, 88, 98, 102, 108] [13, 49, 69, 82, 85, 89, 109, 119] [41, 50, 61, 82, 100, 107, 110, 112] [15, 51, 80, 81, 87, 101, 108, 111] [31, 52, 62, 88, 93, 103, 112, 115] [32, 53, 86, 87, 90, 94, 104, 113] [34, 54, 64, 85, 99, 106, 112, 114] [49, 55, 65, 91, 99, 105, 108, 115] [24, 56, 72, 90, 96, 100, 109, 116] [1, 57, 67, 71, 102, 111, 115, 117] [43, 58, 69, 76, 105, 107, 113, 118] [5, 59, 74, 79, 93, 110, 116, 119]
H_Z (60 checks, sparse supports)
[0, 1, 3, 9, 25, 45, 60, 71] [1, 2, 6, 11, 14, 50, 61, 117] [2, 4, 5, 8, 16, 52, 62, 79] [3, 10, 14, 15, 19, 23, 63, 81] [4, 11, 13, 20, 41, 54, 64, 82] [0, 5, 7, 13, 19, 55, 65, 119] [4, 6, 15, 18, 25, 29, 66, 101] [3, 7, 16, 26, 44, 57, 67, 75] [7, 8, 11, 18, 24, 32, 68, 90] [6, 9, 12, 19, 49, 58, 69, 91] [0, 6, 10, 30, 34, 36, 64, 70] [5, 11, 21, 31, 35, 57, 71, 93] [10, 12, 22, 29, 32, 56, 72, 94] [9, 10, 13, 16, 24, 38, 73, 109] [8, 14, 17, 25, 35, 59, 74, 84] [1, 8, 15, 21, 36, 40, 75, 111] [0, 16, 27, 37, 41, 58, 61, 76] [15, 17, 28, 32, 38, 46, 77, 87] [2, 13, 18, 21, 27, 43, 69, 78] [1, 19, 22, 30, 44, 59, 79, 102] [18, 20, 23, 31, 38, 51, 80, 103] [4, 21, 24, 42, 46, 51, 72, 81] [0, 22, 33, 43, 49, 50, 82, 105] [17, 21, 23, 34, 44, 48, 83, 106] [3, 5, 24, 27, 30, 48, 84, 116] [2, 25, 28, 36, 49, 54, 65, 85] [23, 24, 26, 29, 37, 53, 86, 96] [7, 10, 27, 47, 51, 53, 66, 87] [1, 28, 35, 39, 48, 52, 67, 88] [20, 27, 29, 34, 40, 49, 89, 99] [9, 15, 30, 33, 53, 56, 80, 90] [2, 31, 34, 41, 42, 55, 91, 112] [26, 30, 32, 35, 40, 43, 92, 113] [3, 12, 33, 40, 52, 59, 83, 93] [4, 34, 38, 39, 45, 53, 94, 114] [2, 7, 33, 35, 37, 46, 63, 95] [14, 18, 36, 39, 46, 56, 73, 96] [5, 37, 40, 44, 45, 47, 74, 97] [12, 36, 38, 41, 43, 48, 98, 107] [6, 17, 39, 43, 54, 55, 76, 99] [7, 23, 40, 42, 50, 56, 77, 100] [5, 9, 22, 39, 41, 51, 101, 110] [8, 20, 42, 45, 48, 57, 92, 102] [9, 29, 43, 46, 47, 52, 103, 118] [0, 14, 28, 42, 44, 53, 68, 104] [11, 23, 28, 45, 55, 58, 95, 105] [8, 12, 26, 27, 46, 54, 70, 106] [13, 26, 34, 47, 50, 58, 85, 107] [14, 32, 33, 48, 51, 55, 86, 108] [1, 4, 31, 47, 49, 56, 109, 115] [16, 29, 31, 45, 50, 59, 88, 110] [12, 13, 17, 30, 51, 57, 89, 111] [19, 32, 37, 50, 52, 54, 104, 112] [3, 17, 20, 36, 53, 58, 78, 113] [16, 18, 22, 35, 47, 54, 60, 114] [22, 25, 38, 52, 55, 57, 97, 115] [6, 20, 21, 26, 56, 59, 98, 116] [19, 24, 28, 33, 41, 57, 100, 117] [10, 25, 31, 39, 44, 58, 62, 118] [11, 15, 37, 42, 49, 59, 108, 119]
Code ID 120-6-15 · download JSON · raw on GitHub