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[[120,12,10]] d =
n
120
k
12
d
10
kd²/n
10.0
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 10, d_Z = 10 · 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 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[0, 4, 47, 59, 62, 66, 68, 76, 82, 94]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[60, 61, 62, 65, 68, 73, 78, 84, 90, 96]
certificate exact, d = 10 · CryptoMiniSat 5.14 SAT
X: no logical < 10 exists; Z: no logical < 10 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 4 · H_Z 4 (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)×240 (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): 240 (2,5): 720 (2,6): 240 (3,3): 80 (3,4): 120 (3,5): 1200 (3,6): 6660 (3,7): 7620 (3,8): 1740 (3,9): 720 (3,10): 60
trapping sets H_Z (1,3)×60 (2,4)×240 (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): 240 (2,5): 720 (2,6): 240 (3,3): 80 (3,4): 120 (3,5): 1200 (3,6): 6660 (3,7): 7620 (3,8): 1740 (3,9): 720 (3,10): 60

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

[[120,12,10]] — two-block group-algebra code on SmallGroup(60,1)

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 = 10.00.

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_k12.txt, line 4. This row is SmallGroup(60,1) with nonidentity GAP supports a=[5, 23], b=[2, 10, 26]. 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=12), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 10, not an exact certificate. The published distance (d=10) 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(60,1), 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, 23] and b=[2, 10, 26] 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) = 12. 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, 12, 43, 60, 66, 84, 102] [1, 7, 56, 60, 61, 91, 107] [2, 4, 22, 62, 63, 74, 96] [3, 23, 52, 61, 63, 90, 97] [0, 4, 53, 64, 89, 110, 112] [5, 15, 17, 62, 65, 66, 102] [6, 16, 59, 63, 66, 78, 96] [7, 17, 36, 67, 72, 114, 115] [8, 10, 34, 61, 68, 69, 85] [9, 11, 35, 60, 65, 69, 107] [2, 10, 12, 70, 71, 101, 116] [3, 11, 58, 71, 77, 103, 117] [4, 12, 30, 72, 76, 98, 118] [13, 26, 28, 63, 68, 73, 74] [14, 27, 29, 61, 69, 74, 90] [1, 15, 28, 75, 76, 82, 118] [16, 29, 47, 76, 83, 105, 108] [1, 17, 48, 64, 77, 104, 119] [18, 20, 45, 65, 78, 79, 97] [19, 21, 46, 62, 66, 73, 79] [8, 20, 22, 77, 80, 81, 111] [9, 21, 23, 64, 81, 88, 119] [0, 10, 22, 82, 83, 87, 108] [11, 23, 42, 67, 83, 109, 113] [24, 38, 40, 69, 78, 84, 85] [25, 39, 41, 60, 65, 79, 85] [5, 26, 40, 83, 86, 87, 94] [6, 27, 41, 67, 87, 95, 113] [5, 7, 28, 70, 88, 89, 112] [6, 29, 55, 71, 89, 92, 116] [30, 32, 53, 73, 90, 91, 107] [31, 33, 54, 68, 74, 84, 91] [18, 32, 34, 88, 92, 93, 117] [19, 33, 35, 70, 89, 93, 100] [2, 20, 34, 67, 94, 95, 99] [3, 21, 35, 72, 75, 95, 115] [36, 48, 50, 79, 90, 96, 97] [37, 49, 51, 62, 73, 91, 97] [13, 38, 50, 95, 98, 99, 105] [14, 39, 51, 72, 75, 99, 106] [13, 15, 40, 71, 80, 100, 101] [14, 16, 41, 77, 81, 101, 103] [42, 44, 58, 78, 85, 96, 102] [30, 43, 45, 100, 103, 104, 119] [31, 44, 46, 80, 101, 104, 110] [8, 32, 45, 75, 105, 106, 109] [9, 33, 46, 76, 82, 86, 106] [47, 55, 57, 68, 84, 102, 107] [24, 48, 56, 106, 108, 109, 113] [25, 49, 57, 82, 86, 109, 114] [24, 26, 50, 81, 92, 110, 111] [25, 27, 51, 64, 88, 93, 111] [42, 52, 54, 92, 111, 112, 116] [18, 43, 53, 86, 113, 114, 115] [19, 44, 54, 87, 94, 98, 114] [37, 55, 59, 94, 98, 115, 118] [36, 38, 56, 93, 103, 116, 117] [37, 39, 57, 70, 100, 104, 117] [31, 52, 58, 99, 105, 108, 118] [47, 49, 59, 80, 110, 112, 119]
H_Z (60 checks, sparse supports)
[0, 1, 9, 25, 60, 64, 82] [1, 3, 8, 14, 61, 75, 77] [2, 5, 19, 37, 62, 70, 94] [2, 3, 6, 13, 63, 71, 95] [4, 17, 21, 51, 62, 64, 72] [5, 9, 18, 25, 65, 86, 88] [0, 5, 6, 19, 66, 87, 89] [7, 23, 27, 34, 61, 67, 88] [8, 13, 31, 47, 68, 80, 105] [8, 9, 14, 24, 69, 81, 106] [10, 28, 33, 57, 68, 70, 82] [10, 11, 29, 40, 69, 71, 83] [7, 12, 35, 39, 60, 70, 72] [13, 19, 30, 37, 73, 98, 100] [2, 13, 14, 31, 74, 99, 101] [15, 35, 39, 45, 65, 75, 100] [12, 15, 16, 46, 66, 76, 101] [11, 17, 20, 41, 65, 67, 77] [6, 18, 24, 42, 78, 92, 113] [18, 19, 25, 36, 79, 93, 114] [20, 40, 44, 59, 78, 80, 94] [20, 21, 41, 50, 79, 81, 95] [15, 22, 46, 49, 62, 80, 82] [16, 22, 23, 26, 63, 81, 83] [0, 24, 31, 47, 84, 108, 110] [8, 24, 25, 42, 85, 109, 111] [26, 46, 49, 53, 73, 86, 110] [22, 26, 27, 54, 74, 87, 111] [21, 28, 32, 51, 73, 75, 88] [4, 28, 29, 33, 74, 76, 89] [3, 14, 30, 36, 72, 90, 103] [1, 30, 31, 37, 91, 104, 118] [29, 32, 50, 52, 90, 92, 105] [32, 33, 51, 56, 91, 93, 106] [26, 34, 54, 55, 68, 92, 94] [27, 34, 35, 38, 69, 93, 95] [2, 6, 36, 42, 67, 96, 116] [3, 18, 36, 37, 97, 115, 117] [12, 38, 54, 55, 84, 98, 116] [34, 38, 39, 58, 85, 99, 117] [33, 40, 43, 57, 84, 86, 100] [10, 40, 41, 44, 85, 87, 101] [0, 5, 42, 47, 83, 102, 112] [11, 41, 43, 56, 60, 103, 113] [17, 43, 44, 57, 102, 104, 114] [16, 38, 45, 58, 78, 103, 105] [39, 45, 46, 48, 79, 104, 106] [1, 9, 30, 47, 76, 107, 119] [16, 22, 48, 58, 77, 96, 108] [23, 45, 48, 49, 97, 109, 119] [4, 44, 50, 59, 96, 98, 110] [20, 50, 51, 52, 97, 99, 111] [4, 28, 52, 59, 63, 112, 118] [23, 27, 48, 53, 64, 90, 113] [7, 49, 53, 54, 91, 112, 114] [7, 35, 53, 55, 89, 107, 115] [10, 29, 52, 56, 61, 108, 116] [11, 32, 56, 57, 107, 109, 117] [12, 15, 55, 58, 71, 102, 118] [17, 21, 43, 59, 66, 115, 119]
Code ID 120-12-10 · download JSON · raw on GitHub