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[[120,40,2]] d =
n
120
k
40
d
2
kd²/n
1.333
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[64, 71]
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 7 · H_Z 7
qubit degrees H_X 2–5 (mean 3.5) · H_Z 2–5 (mean 3.5)
trapping sets H_X (1,2)×60 (2,0)×30 (3,2)×240 (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 (1,5): 60 (2,0): 30 (2,3): 120 (2,4): 60 (2,5): 360 (2,6): 30 (2,8): 360 (3,2): 240 (3,3): 720 (3,4): 180 (3,5): 1440 (3,6): 1080 (3,7): 180 (3,8): 4740 (3,9): 380 (3,11): 3060
trapping sets H_Z (1,2)×60 (2,0)×30 (3,2)×240 (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 (1,5): 60 (2,0): 30 (2,3): 120 (2,4): 60 (2,5): 360 (2,6): 30 (2,8): 360 (3,2): 240 (3,3): 720 (3,4): 180 (3,5): 1440 (3,6): 1080 (3,7): 180 (3,8): 4740 (3,9): 380 (3,11): 3060

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_wt7wtL2_order60_k40.txt, line 1. nonidentity supports a=[4], b=[4, 6, 16, 29].
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

[[120,40,2]] — 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 = 1.33.

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_wt7wtL2_order60_k40.txt, line 1. This row is SmallGroup(60,1) with nonidentity GAP supports a=[4], b=[4, 6, 16, 29]. 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=40), 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 (d=2) 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=[4, 6, 16, 29] 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) = 40. 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, 3, 60, 63, 107, 115, 119] [1, 6, 61, 66, 90, 103, 113] [2, 9, 62, 66, 69, 76, 89] [0, 3, 60, 63, 96, 108, 116] [4, 11, 64, 71, 107, 115, 119] [5, 14, 60, 64, 65, 72, 74] [1, 6, 61, 66, 102, 112, 118] [7, 16, 67, 76, 90, 103, 113] [8, 19, 68, 74, 79, 87, 101] [2, 9, 61, 62, 67, 69, 77] [10, 21, 66, 70, 76, 81, 89] [4, 11, 64, 71, 96, 108, 116] [12, 23, 72, 83, 107, 115, 119] [13, 25, 62, 70, 73, 82, 85] [5, 14, 63, 65, 71, 74, 83] [15, 27, 60, 64, 72, 75, 87] [7, 16, 67, 76, 102, 112, 118] [17, 29, 77, 89, 90, 103, 113] [18, 31, 78, 85, 91, 99, 111] [8, 19, 65, 68, 75, 79, 88] [20, 33, 74, 80, 87, 93, 101] [10, 21, 61, 67, 70, 77, 81] [22, 35, 66, 76, 82, 89, 95] [12, 23, 72, 83, 96, 108, 116] [24, 37, 68, 80, 84, 94, 97] [13, 25, 69, 73, 81, 85, 95] [26, 39, 62, 70, 82, 86, 99] [15, 27, 63, 71, 75, 83, 87] [28, 41, 60, 64, 72, 88, 101] [17, 29, 77, 89, 102, 112, 118] [30, 42, 90, 97, 102, 109, 117] [18, 31, 73, 78, 86, 91, 100] [32, 44, 85, 92, 99, 104, 111] [20, 33, 65, 75, 80, 88, 93] [34, 46, 74, 87, 94, 101, 106] [22, 35, 61, 67, 77, 82, 95] [36, 47, 78, 92, 96, 105, 107] [24, 37, 79, 84, 93, 97, 106] [38, 49, 68, 80, 94, 98, 109] [26, 39, 69, 81, 86, 95, 99] [40, 51, 62, 70, 82, 100, 111] [28, 41, 63, 71, 83, 88, 101] [30, 42, 84, 90, 98, 102, 110] [43, 52, 97, 103, 109, 112, 117] [32, 44, 73, 86, 92, 100, 104] [45, 54, 85, 99, 105, 111, 114] [34, 46, 65, 75, 88, 94, 106] [36, 47, 91, 96, 104, 107, 114] [48, 55, 78, 92, 105, 108, 115] [38, 49, 79, 93, 98, 106, 109] [50, 57, 68, 80, 94, 110, 117] [40, 51, 69, 81, 95, 100, 111] [43, 52, 84, 98, 103, 110, 112] [53, 58, 97, 109, 113, 117, 118] [45, 54, 73, 86, 100, 105, 114] [48, 55, 91, 104, 108, 114, 115] [56, 59, 78, 92, 105, 116, 119] [50, 57, 79, 93, 106, 110, 117] [53, 58, 84, 98, 110, 113, 118] [56, 59, 91, 104, 114, 116, 119]
H_Z (60 checks, sparse supports)
[0, 3, 5, 15, 28, 60, 63] [1, 6, 9, 21, 35, 61, 66] [2, 9, 13, 26, 40, 62, 69] [0, 3, 14, 27, 41, 60, 63] [4, 5, 11, 15, 28, 64, 71] [5, 14, 19, 33, 46, 65, 74] [1, 2, 6, 10, 22, 61, 66] [7, 9, 16, 21, 35, 67, 76] [8, 19, 24, 38, 50, 68, 79] [2, 9, 25, 39, 51, 62, 69] [10, 13, 21, 26, 40, 70, 81] [4, 11, 14, 27, 41, 64, 71] [5, 12, 15, 23, 28, 72, 83] [13, 25, 31, 44, 54, 73, 85] [5, 8, 14, 20, 34, 65, 74] [15, 19, 27, 33, 46, 75, 87] [2, 7, 10, 16, 22, 67, 76] [9, 17, 21, 29, 35, 77, 89] [18, 31, 36, 48, 56, 78, 91] [8, 19, 37, 49, 57, 68, 79] [20, 24, 33, 38, 50, 80, 93] [10, 21, 25, 39, 51, 70, 81] [13, 22, 26, 35, 40, 82, 95] [12, 14, 23, 27, 41, 72, 83] [24, 37, 42, 52, 58, 84, 97] [13, 18, 25, 32, 45, 73, 85] [26, 31, 39, 44, 54, 86, 99] [8, 15, 20, 27, 34, 75, 87] [19, 28, 33, 41, 46, 88, 101] [2, 10, 17, 22, 29, 77, 89] [1, 7, 17, 30, 42, 90, 102] [18, 31, 47, 55, 59, 78, 91] [32, 36, 44, 48, 56, 92, 104] [20, 33, 37, 49, 57, 80, 93] [24, 34, 38, 46, 50, 94, 106] [22, 25, 35, 39, 51, 82, 95] [3, 11, 23, 36, 47, 96, 107] [24, 30, 37, 43, 53, 84, 97] [38, 42, 49, 52, 58, 98, 109] [18, 26, 32, 39, 45, 86, 99] [31, 40, 44, 51, 54, 100, 111] [8, 20, 28, 34, 41, 88, 101] [6, 16, 29, 30, 42, 90, 102] [1, 7, 17, 43, 52, 103, 112] [32, 44, 47, 55, 59, 92, 104] [36, 45, 48, 54, 56, 105, 114] [34, 37, 46, 49, 57, 94, 106] [0, 4, 12, 36, 47, 96, 107] [3, 11, 23, 48, 55, 108, 115] [30, 38, 43, 49, 53, 98, 109] [42, 50, 52, 57, 58, 110, 117] [18, 32, 40, 45, 51, 100, 111] [6, 16, 29, 43, 52, 103, 112] [1, 7, 17, 53, 58, 113, 118] [45, 47, 54, 55, 59, 105, 114] [0, 4, 12, 48, 55, 108, 115] [3, 11, 23, 56, 59, 116, 119] [30, 43, 50, 53, 57, 110, 117] [6, 16, 29, 53, 58, 113, 118] [0, 4, 12, 56, 59, 116, 119]
Code ID 120-40-2 · download JSON · raw on GitHub