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[[120,60,2]] d =
n
120
k
60
d
2
kd²/n
2.0
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[110, 117]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[17, 29]
certificate exact, d = 2 · scipy/HiGHS MILP
X: no logical < 2 exists; Z: no logical < 2 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 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 (2,0)×60 (3,2)×1140 (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 (2,0): 60 (2,4): 420 (2,8): 240 (3,2): 1140 (3,6): 3120 (3,10): 1920
trapping sets H_Z (1,2)×60 (2,0)×60 (3,2)×1140 (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 (2,0): 60 (2,4): 420 (2,8): 240 (3,2): 1140 (3,6): 3120 (3,10): 1920

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(60,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order60_k60.txt, line 1. nonidentity supports a=[4], b=[2, 4, 7, 11, 22].
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

[[120,60,2]] — two-block group-algebra code on SmallGroup(60,1)

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 = 2.00.

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_wt8wtL2_order60_k60.txt, line 1. This row is SmallGroup(60,1) with nonidentity GAP supports a=[4], b=[2, 4, 7, 11, 22]. 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=60), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 2, 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(60,1), 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=[4], b=[2, 4, 7, 11, 22] 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) = 60. 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, 3, 60, 61, 63, 66, 113, 118] [1, 6, 60, 61, 63, 66, 116, 119] [2, 9, 62, 65, 69, 72, 74, 83] [0, 3, 60, 61, 63, 66, 113, 118] [4, 11, 64, 71, 77, 89, 90, 102] [5, 14, 62, 65, 69, 74, 77, 89] [1, 6, 60, 61, 63, 66, 116, 119] [7, 16, 67, 72, 76, 83, 96, 107] [8, 19, 68, 73, 79, 82, 85, 95] [2, 9, 62, 65, 69, 72, 74, 83] [10, 21, 60, 63, 70, 81, 88, 101] [4, 11, 64, 71, 77, 89, 90, 102] [12, 23, 67, 72, 76, 83, 103, 112] [13, 25, 68, 73, 79, 85, 88, 101] [5, 14, 62, 65, 69, 74, 77, 89] [15, 27, 61, 66, 75, 82, 87, 95] [7, 16, 67, 72, 76, 83, 96, 107] [17, 29, 64, 71, 77, 89, 108, 115] [18, 31, 78, 84, 91, 94, 97, 106] [8, 19, 68, 73, 79, 82, 85, 95] [20, 33, 62, 69, 80, 93, 100, 111] [10, 21, 60, 63, 70, 81, 88, 101] [22, 35, 64, 71, 75, 82, 87, 95] [12, 23, 67, 72, 76, 83, 103, 112] [24, 37, 78, 84, 91, 97, 100, 111] [13, 25, 68, 73, 79, 85, 88, 101] [26, 39, 65, 74, 86, 94, 99, 106] [15, 27, 61, 66, 75, 82, 87, 95] [28, 41, 67, 70, 76, 81, 88, 101] [17, 29, 64, 71, 77, 89, 108, 115] [30, 42, 90, 96, 102, 105, 107, 114] [18, 31, 78, 84, 91, 94, 97, 106] [32, 44, 68, 79, 92, 104, 110, 117] [20, 33, 62, 69, 80, 93, 100, 111] [34, 46, 70, 81, 86, 94, 99, 106] [22, 35, 64, 71, 75, 82, 87, 95] [36, 47, 90, 96, 102, 107, 110, 117] [24, 37, 78, 84, 91, 97, 100, 111] [38, 49, 73, 85, 98, 105, 109, 114] [26, 39, 65, 74, 86, 94, 99, 106] [40, 51, 75, 80, 87, 93, 100, 111] [28, 41, 67, 70, 76, 81, 88, 101] [30, 42, 90, 96, 102, 105, 107, 114] [43, 52, 78, 91, 103, 112, 116, 119] [32, 44, 68, 79, 92, 104, 110, 117] [45, 54, 80, 93, 98, 105, 109, 114] [34, 46, 70, 81, 86, 94, 99, 106] [36, 47, 90, 96, 102, 107, 110, 117] [48, 55, 84, 97, 108, 113, 115, 118] [38, 49, 73, 85, 98, 105, 109, 114] [50, 57, 86, 92, 99, 104, 110, 117] [40, 51, 75, 80, 87, 93, 100, 111] [43, 52, 78, 91, 103, 112, 116, 119] [53, 58, 92, 104, 108, 113, 115, 118] [45, 54, 80, 93, 98, 105, 109, 114] [48, 55, 84, 97, 108, 113, 115, 118] [56, 59, 98, 103, 109, 112, 116, 119] [50, 57, 86, 92, 99, 104, 110, 117] [53, 58, 92, 104, 108, 113, 115, 118] [56, 59, 98, 103, 109, 112, 116, 119]
H_Z (60 checks, sparse supports)
[0, 1, 3, 6, 10, 21, 60, 63] [0, 1, 3, 6, 15, 27, 61, 66] [2, 5, 9, 14, 20, 33, 62, 69] [0, 1, 3, 6, 10, 21, 60, 63] [4, 11, 17, 22, 29, 35, 64, 71] [2, 5, 9, 14, 26, 39, 65, 74] [0, 1, 3, 6, 15, 27, 61, 66] [7, 12, 16, 23, 28, 41, 67, 76] [8, 13, 19, 25, 32, 44, 68, 79] [2, 5, 9, 14, 20, 33, 62, 69] [10, 21, 28, 34, 41, 46, 70, 81] [4, 11, 17, 22, 29, 35, 64, 71] [2, 7, 9, 12, 16, 23, 72, 83] [8, 13, 19, 25, 38, 49, 73, 85] [2, 5, 9, 14, 26, 39, 65, 74] [15, 22, 27, 35, 40, 51, 75, 87] [7, 12, 16, 23, 28, 41, 67, 76] [4, 5, 11, 14, 17, 29, 77, 89] [18, 24, 31, 37, 43, 52, 78, 91] [8, 13, 19, 25, 32, 44, 68, 79] [20, 33, 40, 45, 51, 54, 80, 93] [10, 21, 28, 34, 41, 46, 70, 81] [8, 15, 19, 22, 27, 35, 82, 95] [2, 7, 9, 12, 16, 23, 72, 83] [18, 24, 31, 37, 48, 55, 84, 97] [8, 13, 19, 25, 38, 49, 73, 85] [26, 34, 39, 46, 50, 57, 86, 99] [15, 22, 27, 35, 40, 51, 75, 87] [10, 13, 21, 25, 28, 41, 88, 101] [4, 5, 11, 14, 17, 29, 77, 89] [4, 11, 30, 36, 42, 47, 90, 102] [18, 24, 31, 37, 43, 52, 78, 91] [32, 44, 50, 53, 57, 58, 92, 104] [20, 33, 40, 45, 51, 54, 80, 93] [18, 26, 31, 34, 39, 46, 94, 106] [8, 15, 19, 22, 27, 35, 82, 95] [7, 16, 30, 36, 42, 47, 96, 107] [18, 24, 31, 37, 48, 55, 84, 97] [38, 45, 49, 54, 56, 59, 98, 109] [26, 34, 39, 46, 50, 57, 86, 99] [20, 24, 33, 37, 40, 51, 100, 111] [10, 13, 21, 25, 28, 41, 88, 101] [4, 11, 30, 36, 42, 47, 90, 102] [12, 23, 43, 52, 56, 59, 103, 112] [32, 44, 50, 53, 57, 58, 92, 104] [30, 38, 42, 45, 49, 54, 105, 114] [18, 26, 31, 34, 39, 46, 94, 106] [7, 16, 30, 36, 42, 47, 96, 107] [17, 29, 48, 53, 55, 58, 108, 115] [38, 45, 49, 54, 56, 59, 98, 109] [32, 36, 44, 47, 50, 57, 110, 117] [20, 24, 33, 37, 40, 51, 100, 111] [12, 23, 43, 52, 56, 59, 103, 112] [0, 3, 48, 53, 55, 58, 113, 118] [30, 38, 42, 45, 49, 54, 105, 114] [17, 29, 48, 53, 55, 58, 108, 115] [1, 6, 43, 52, 56, 59, 116, 119] [32, 36, 44, 47, 50, 57, 110, 117] [0, 3, 48, 53, 55, 58, 113, 118] [1, 6, 43, 52, 56, 59, 116, 119]
Code ID 120-60-2 · download JSON · raw on GitHub