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[[120,16,8]] d =
n
120
k
16
d
8
kd²/n
8.533
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 8, d_Z = 8 · 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 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[66, 84, 88, 100, 107, 109, 111, 117]
d_Z 8 · witness weight 8 (claimed upper_bound)
witness operator (support, 8 qubits)
[3, 9, 11, 22, 30, 34, 42, 44]
certificate exact, d = 8 · CryptoMiniSat 5.14 SAT
X: no logical < 8 exists; Z: no logical < 8 exists

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): 120 (3,5): 780 (3,6): 6020 (3,7): 8700 (3,8): 3180 (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): 120 (3,5): 780 (3,6): 6020 (3,7): 8700 (3,8): 3180 (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

[[120,16,8]] — 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 = 8.53.

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_k16.txt, line 6. This row is SmallGroup(60,2) with nonidentity GAP supports a=[5, 34], b=[2, 6, 37]. 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=16), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 8, not an exact certificate. The published distance (d=8) 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, 34] and b=[2, 6, 37] 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) = 16. 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, 31, 35, 60, 66, 75, 84] [1, 7, 48, 60, 61, 68, 80] [2, 12, 45, 62, 66, 74, 85] [3, 42, 46, 61, 63, 73, 86] [0, 4, 43, 64, 65, 111, 119] [5, 15, 29, 60, 62, 65, 90] [6, 16, 55, 63, 66, 70, 78] [7, 17, 56, 67, 69, 95, 113] [8, 21, 53, 67, 68, 74, 84] [9, 22, 54, 61, 65, 69, 96] [2, 10, 23, 70, 73, 111, 117] [3, 11, 52, 71, 74, 101, 116] [4, 12, 53, 72, 100, 110, 115] [13, 25, 39, 62, 68, 71, 73] [14, 26, 40, 63, 69, 74, 79] [15, 27, 41, 75, 78, 95, 105] [16, 28, 59, 62, 76, 106, 118] [13, 17, 29, 77, 83, 103, 114] [18, 32, 58, 65, 73, 76, 78] [8, 19, 33, 61, 79, 117, 119] [9, 20, 34, 80, 84, 101, 110] [10, 21, 35, 81, 100, 109, 116] [11, 22, 58, 82, 89, 108, 117] [8, 12, 23, 83, 88, 99, 107] [24, 36, 49, 64, 69, 78, 84] [25, 37, 50, 63, 85, 105, 113] [26, 38, 51, 68, 86, 106, 114] [1, 27, 39, 83, 87, 93, 118] [24, 28, 40, 88, 94, 105, 112] [25, 29, 41, 72, 89, 91, 104] [18, 30, 44, 66, 90, 110, 116] [19, 31, 45, 91, 101, 109, 115] [20, 32, 46, 89, 92, 99, 119] [0, 21, 33, 88, 93, 98, 108] [18, 22, 34, 77, 94, 97, 109] [19, 23, 35, 76, 87, 95, 96] [36, 47, 57, 60, 96, 114, 118] [5, 37, 48, 93, 97, 103, 106] [6, 38, 49, 94, 98, 104, 113] [7, 39, 50, 72, 81, 99, 112] [36, 40, 51, 82, 93, 100, 102] [1, 37, 41, 64, 79, 92, 101] [30, 42, 54, 99, 102, 108, 111] [2, 31, 43, 89, 98, 103, 107] [3, 32, 44, 77, 87, 104, 115] [4, 33, 45, 76, 86, 97, 105] [30, 34, 46, 67, 85, 98, 106] [14, 47, 55, 95, 104, 107, 112] [15, 48, 56, 81, 91, 94, 108] [16, 49, 57, 82, 92, 103, 109] [5, 17, 50, 64, 70, 102, 110] [6, 47, 51, 71, 81, 90, 111] [9, 42, 52, 87, 97, 100, 112] [10, 43, 53, 77, 86, 96, 113] [11, 44, 54, 67, 75, 107, 114] [26, 55, 59, 83, 92, 102, 115] [13, 27, 56, 70, 79, 82, 116] [14, 28, 57, 71, 80, 91, 117] [20, 52, 58, 75, 85, 88, 118] [24, 38, 59, 72, 80, 90, 119]
H_Z (60 checks, sparse supports)
[0, 1, 5, 36, 60, 64, 93] [1, 3, 9, 19, 61, 87, 101] [2, 5, 13, 16, 62, 70, 103] [3, 6, 14, 25, 63, 71, 104] [4, 24, 41, 50, 64, 72, 105] [4, 5, 9, 18, 65, 97, 110] [0, 2, 6, 30, 66, 98, 111] [7, 8, 46, 54, 61, 67, 99] [1, 8, 13, 26, 68, 79, 83] [7, 9, 14, 24, 69, 80, 112] [6, 10, 50, 56, 70, 81, 113] [11, 13, 51, 57, 71, 82, 114] [12, 29, 39, 59, 62, 72, 83] [3, 10, 13, 18, 73, 77, 116] [2, 8, 11, 14, 74, 107, 117] [0, 15, 54, 58, 65, 75, 108] [16, 18, 35, 45, 66, 76, 109] [17, 34, 44, 53, 67, 77, 110] [6, 15, 18, 24, 78, 90, 94] [14, 19, 41, 56, 79, 91, 95] [1, 20, 57, 59, 80, 92, 118] [21, 39, 48, 51, 68, 81, 93] [22, 40, 49, 56, 69, 82, 94] [17, 23, 27, 55, 70, 83, 95] [0, 8, 20, 24, 84, 88, 119] [2, 25, 46, 58, 73, 85, 89] [3, 26, 45, 53, 74, 86, 115] [27, 35, 44, 52, 75, 87, 116] [23, 28, 33, 58, 76, 88, 117] [22, 29, 32, 43, 65, 77, 89] [5, 30, 51, 59, 90, 102, 106] [29, 31, 48, 57, 60, 91, 103] [32, 41, 49, 55, 78, 92, 104] [27, 33, 37, 40, 79, 93, 105] [28, 34, 38, 48, 80, 94, 106] [7, 15, 35, 47, 60, 81, 95] [9, 35, 36, 53, 84, 96, 100] [34, 37, 45, 52, 85, 97, 101] [33, 38, 43, 46, 86, 98, 119] [23, 32, 39, 42, 73, 87, 99] [12, 21, 40, 52, 74, 88, 100] [11, 20, 31, 41, 75, 89, 101] [40, 42, 50, 55, 63, 102, 112] [17, 37, 43, 49, 64, 103, 113] [29, 38, 44, 47, 90, 104, 114] [15, 25, 28, 45, 62, 91, 105] [16, 26, 37, 46, 63, 92, 106] [23, 43, 47, 54, 96, 107, 111] [22, 33, 42, 48, 61, 97, 108] [21, 31, 34, 49, 84, 98, 109] [12, 20, 30, 50, 85, 99, 110] [4, 10, 42, 51, 86, 100, 111] [28, 39, 47, 52, 71, 112, 118] [7, 25, 38, 53, 68, 72, 113] [17, 26, 36, 54, 69, 102, 114] [12, 31, 44, 55, 66, 107, 115] [11, 21, 30, 56, 67, 108, 116] [10, 19, 22, 57, 96, 109, 117] [16, 27, 36, 58, 78, 82, 118] [4, 19, 32, 59, 76, 115, 119]
Code ID 120-16-8 · download JSON · raw on GitHub