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[[120,4,15]] d ≤
n
120
k
4
d
15
kd²/n
7.5
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[19, 24, 31, 45, 51, 60, 63, 68, 79, 89, 91, 99, 106, 108, 118]
d_Z 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[2, 16, 22, 23, 25, 28, 33, 42, 68, 71, 88, 95, 104, 106, 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 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)×80 (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): 80 (3,4): 120 (3,5): 840 (3,6): 6000 (3,7): 8340 (3,8): 3240 (3,9): 1080 (3,10): 240
trapping sets H_Z (1,3)×60 (2,4)×180 (3,3)×80 (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): 80 (3,4): 120 (3,5): 840 (3,6): 6000 (3,7): 8340 (3,8): 3240 (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

[[120,4,15]] — 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 = 4·225/120 = 7.50.

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_k4.txt, line 10. This row is SmallGroup(60,2) with nonidentity GAP supports a=[3,31], b=[2,11,46]. 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=4), 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 (d=15) 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=[3,31] and b=[2,11,46] 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) = 4. 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, 8, 54, 60, 66, 79, 113] [1, 13, 26, 60, 61, 85, 116] [0, 2, 58, 62, 64, 74, 95] [3, 18, 45, 61, 63, 90, 118] [4, 9, 19, 64, 68, 91, 111] [1, 5, 36, 62, 65, 67, 101] [6, 15, 24, 63, 66, 96, 119] [7, 25, 38, 67, 73, 95, 97] [2, 8, 46, 68, 70, 84, 105] [3, 9, 53, 65, 69, 71, 106] [4, 10, 18, 60, 70, 72, 117] [2, 11, 30, 71, 78, 101, 102] [12, 20, 31, 72, 79, 100, 103] [5, 13, 16, 68, 73, 75, 110] [6, 14, 25, 69, 74, 76, 111] [7, 15, 47, 61, 75, 77, 105] [16, 27, 36, 76, 84, 106, 107] [17, 37, 49, 77, 83, 85, 108] [9, 18, 35, 73, 78, 80, 114] [3, 10, 19, 62, 79, 81, 119] [8, 11, 20, 63, 80, 82, 110] [12, 21, 30, 64, 81, 83, 109] [10, 22, 42, 82, 89, 90, 112] [23, 32, 43, 83, 88, 91, 113] [7, 14, 24, 78, 84, 86, 117] [15, 25, 28, 65, 85, 87, 113] [16, 26, 37, 66, 86, 88, 114] [17, 27, 55, 67, 87, 89, 93] [28, 39, 47, 88, 94, 96, 115] [29, 48, 57, 72, 89, 97, 116] [0, 20, 30, 69, 90, 92, 116] [11, 21, 31, 70, 91, 93, 115] [19, 22, 32, 71, 92, 94, 99] [23, 33, 42, 72, 93, 95, 98] [21, 34, 52, 77, 94, 102, 118] [35, 44, 53, 68, 76, 95, 103] [17, 26, 36, 74, 96, 98, 118] [27, 37, 40, 75, 97, 99, 103] [28, 38, 48, 76, 98, 100, 104] [29, 39, 59, 77, 81, 99, 101] [40, 50, 55, 82, 100, 107, 119] [14, 41, 56, 64, 73, 101, 108] [4, 32, 42, 80, 102, 104, 108] [22, 33, 43, 81, 103, 105, 107] [31, 34, 44, 82, 87, 104, 106] [35, 45, 52, 60, 83, 86, 105] [33, 46, 58, 67, 78, 106, 112] [29, 38, 47, 86, 107, 109, 112] [39, 48, 51, 87, 91, 108, 110] [40, 49, 56, 88, 92, 109, 111] [24, 41, 50, 61, 70, 89, 110] [5, 51, 59, 71, 84, 111, 115] [12, 44, 52, 92, 97, 112, 114] [34, 45, 53, 62, 93, 96, 113] [43, 46, 54, 63, 75, 94, 114] [41, 49, 55, 98, 102, 115, 117] [6, 50, 56, 65, 79, 99, 116] [13, 51, 57, 66, 80, 100, 117] [23, 54, 58, 69, 85, 104, 118] [1, 57, 59, 74, 90, 109, 119]
H_Z (60 checks, sparse supports)
[0, 1, 10, 45, 60, 62, 90] [1, 3, 15, 50, 61, 65, 119] [2, 5, 19, 53, 62, 68, 71] [3, 6, 20, 54, 63, 69, 79] [2, 4, 21, 41, 64, 70, 102] [5, 9, 25, 56, 65, 73, 111] [0, 6, 26, 57, 66, 74, 116] [5, 7, 27, 46, 67, 75, 84] [4, 8, 13, 35, 60, 68, 80] [9, 14, 30, 58, 64, 69, 78] [8, 10, 31, 50, 70, 79, 82] [9, 11, 32, 51, 71, 80, 91] [10, 12, 29, 33, 72, 81, 112] [7, 13, 18, 41, 61, 73, 117] [2, 14, 36, 59, 74, 84, 101] [13, 15, 37, 54, 66, 75, 85] [14, 16, 35, 38, 73, 76, 86] [15, 17, 34, 39, 77, 87, 96] [11, 18, 24, 46, 63, 70, 78] [0, 12, 19, 56, 64, 79, 92] [18, 20, 42, 57, 72, 80, 90] [19, 21, 39, 43, 81, 91, 94] [20, 22, 40, 44, 82, 92, 103] [17, 21, 23, 45, 83, 93, 118] [8, 16, 24, 51, 66, 84, 110] [1, 17, 25, 58, 67, 74, 85] [24, 26, 45, 47, 61, 86, 96] [25, 27, 44, 48, 76, 87, 97] [23, 26, 28, 49, 85, 88, 98] [22, 27, 29, 50, 89, 99, 107] [3, 22, 30, 59, 71, 81, 90] [4, 23, 31, 48, 72, 91, 104] [30, 32, 49, 52, 83, 92, 102] [27, 31, 33, 53, 93, 103, 106] [28, 32, 34, 54, 94, 104, 113] [2, 7, 33, 35, 78, 95, 105] [6, 28, 36, 53, 65, 76, 96] [7, 29, 37, 52, 77, 86, 97] [33, 36, 38, 55, 67, 98, 107] [32, 37, 39, 56, 88, 99, 108] [12, 38, 40, 57, 97, 100, 109] [5, 11, 39, 41, 101, 110, 115] [11, 34, 42, 55, 82, 93, 102] [12, 35, 37, 43, 83, 103, 114] [38, 42, 44, 58, 95, 104, 112] [8, 15, 43, 45, 63, 105, 113] [9, 16, 44, 46, 68, 106, 114] [16, 40, 43, 47, 75, 88, 107] [17, 41, 42, 48, 89, 98, 108] [21, 47, 49, 59, 77, 109, 115] [13, 20, 48, 50, 100, 110, 116] [4, 14, 49, 51, 108, 111, 117] [22, 46, 47, 52, 94, 105, 112] [0, 23, 25, 53, 69, 95, 113] [18, 26, 52, 54, 60, 114, 118] [28, 31, 51, 55, 87, 100, 115] [1, 29, 30, 56, 101, 109, 116] [10, 24, 55, 57, 89, 117, 119] [3, 34, 36, 58, 62, 106, 118] [6, 19, 40, 59, 99, 111, 119]
Code ID 120-4-15 · download JSON · raw on GitHub