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[[120,42,3]] d =
n
120
k
42
d
3
kd²/n
3.15
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · 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 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[73, 86, 100]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[9, 21, 35]
certificate exact, d = 3 · scipy/HiGHS MILP
X: no logical < 3 exists; Z: no logical < 3 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 4 · H_Z 4
trapping sets H_X (1,4)×120 (2,4)×480 (3,0)×40 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 120 (2,4): 480 (2,6): 720 (3,0): 40 (3,4): 1800 (3,6): 6240 (3,8): 6000
trapping sets H_Z (1,4)×120 (2,4)×480 (3,0)×40 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 120 (2,4): 480 (2,6): 720 (3,0): 40 (3,4): 1800 (3,6): 6240 (3,8): 6000

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_wt8wtL4_order60_k42.txt, line 2. nonidentity supports a=[2, 5, 8], b=[5, 6, 16].
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

[[120,42,3]] — 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 = 3.15.

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_wt8wtL4_order60_k42.txt, line 2. This row is SmallGroup(60,1) with nonidentity GAP supports a=[2, 5, 8], b=[5, 6, 16]. 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=42), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 3, 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=[2, 5, 8], b=[5, 6, 16] 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) = 42. 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, 6, 12, 16, 60, 72, 107, 115] [0, 1, 7, 12, 61, 77, 90, 103] [2, 14, 22, 27, 62, 66, 76, 82] [1, 3, 7, 23, 63, 83, 96, 108] [0, 4, 16, 29, 60, 64, 107, 119] [2, 5, 15, 22, 60, 64, 65, 88] [3, 6, 16, 23, 66, 89, 102, 112] [0, 4, 7, 17, 61, 67, 90, 113] [8, 25, 34, 39, 68, 74, 87, 94] [5, 9, 15, 35, 61, 67, 69, 95] [2, 10, 27, 41, 62, 66, 70, 89] [3, 7, 11, 17, 63, 71, 96, 116] [4, 6, 12, 29, 64, 72, 115, 119] [8, 13, 26, 34, 62, 70, 73, 100] [9, 14, 27, 35, 63, 71, 74, 101] [2, 10, 15, 28, 60, 65, 72, 75] [3, 11, 16, 29, 66, 76, 102, 118] [1, 4, 12, 17, 67, 77, 103, 113] [18, 37, 45, 49, 78, 85, 99, 105] [13, 19, 26, 46, 65, 75, 79, 106] [8, 20, 39, 51, 68, 74, 80, 101] [9, 15, 21, 28, 61, 69, 77, 81] [10, 14, 22, 41, 70, 76, 82, 89] [1, 11, 17, 23, 71, 83, 108, 116] [18, 24, 38, 45, 68, 80, 84, 110] [19, 25, 39, 46, 69, 81, 85, 111] [8, 20, 26, 40, 62, 73, 82, 86] [9, 21, 27, 41, 63, 74, 83, 87] [5, 10, 22, 28, 64, 72, 75, 88] [6, 11, 23, 29, 76, 89, 112, 118] [30, 47, 53, 55, 90, 97, 109, 113] [24, 31, 38, 54, 73, 86, 91, 114] [18, 32, 49, 57, 78, 85, 92, 111] [19, 26, 33, 40, 65, 79, 88, 93] [20, 25, 34, 51, 80, 87, 94, 101] [5, 21, 28, 35, 67, 77, 81, 95] [30, 36, 48, 53, 78, 92, 96, 116] [31, 37, 49, 54, 79, 93, 97, 117] [18, 32, 38, 50, 68, 84, 94, 98] [19, 33, 39, 51, 69, 85, 95, 99] [13, 20, 34, 40, 70, 82, 86, 100] [14, 21, 35, 41, 71, 83, 87, 101] [36, 42, 48, 58, 84, 98, 102, 118] [30, 43, 55, 59, 90, 97, 103, 117] [31, 38, 44, 50, 73, 91, 100, 104] [32, 37, 45, 57, 92, 99, 105, 111] [13, 33, 40, 46, 75, 88, 93, 106] [42, 47, 55, 58, 91, 104, 107, 119] [30, 43, 48, 56, 78, 96, 105, 108] [31, 44, 49, 57, 79, 97, 106, 109] [24, 32, 45, 50, 80, 94, 98, 110] [25, 33, 46, 51, 81, 95, 99, 111] [42, 48, 52, 56, 84, 102, 110, 112] [43, 47, 53, 59, 103, 109, 113, 117] [24, 44, 50, 54, 86, 100, 104, 114] [42, 52, 55, 59, 91, 107, 114, 115] [36, 43, 53, 56, 92, 105, 108, 116] [37, 44, 54, 57, 93, 106, 109, 117] [36, 52, 56, 58, 98, 110, 112, 118] [47, 52, 58, 59, 104, 114, 115, 119]
H_Z (60 checks, sparse supports)
[0, 4, 5, 15, 60, 61, 64, 67] [1, 7, 9, 21, 61, 63, 77, 83] [2, 10, 13, 26, 62, 65, 70, 75] [3, 11, 14, 27, 63, 66, 71, 76] [4, 5, 12, 28, 64, 67, 72, 77] [5, 15, 19, 33, 65, 69, 88, 95] [2, 6, 10, 16, 60, 66, 72, 89] [7, 9, 17, 35, 61, 63, 67, 71] [8, 20, 24, 38, 68, 73, 80, 86] [9, 21, 25, 39, 69, 74, 81, 87] [10, 13, 22, 40, 70, 75, 82, 88] [11, 14, 23, 41, 71, 76, 83, 89] [0, 12, 15, 28, 60, 61, 72, 77] [13, 26, 31, 44, 73, 79, 100, 106] [8, 14, 20, 27, 62, 74, 82, 101] [15, 19, 28, 46, 65, 69, 75, 81] [2, 16, 22, 29, 60, 64, 66, 76] [1, 17, 21, 35, 67, 71, 77, 83] [18, 32, 36, 48, 78, 84, 92, 98] [19, 33, 37, 49, 79, 85, 93, 99] [20, 24, 34, 50, 80, 86, 94, 100] [21, 25, 35, 51, 81, 87, 95, 101] [2, 22, 26, 40, 62, 65, 82, 88] [3, 23, 27, 41, 63, 66, 83, 89] [24, 38, 42, 52, 84, 91, 110, 114] [18, 25, 32, 39, 68, 85, 94, 111] [26, 31, 40, 54, 73, 79, 86, 93] [8, 27, 34, 41, 62, 70, 74, 87] [5, 28, 33, 46, 75, 81, 88, 95] [6, 10, 22, 29, 64, 72, 76, 89] [1, 7, 30, 43, 90, 96, 103, 108] [31, 44, 47, 55, 91, 97, 104, 109] [32, 36, 45, 56, 92, 98, 105, 110] [33, 37, 46, 57, 93, 99, 106, 111] [8, 34, 38, 50, 68, 73, 94, 100] [9, 35, 39, 51, 69, 74, 95, 101] [3, 11, 36, 48, 96, 102, 116, 118] [30, 37, 43, 49, 78, 97, 105, 117] [38, 42, 50, 58, 84, 91, 98, 104] [18, 39, 45, 51, 68, 80, 85, 99] [13, 40, 44, 54, 86, 93, 100, 106] [14, 20, 34, 41, 70, 82, 87, 101] [6, 16, 42, 52, 102, 107, 112, 115] [1, 17, 43, 53, 103, 108, 113, 116] [44, 47, 54, 59, 104, 109, 114, 117] [18, 45, 48, 56, 78, 84, 105, 110] [19, 46, 49, 57, 79, 85, 106, 111] [0, 4, 47, 55, 90, 107, 113, 119] [3, 23, 48, 56, 96, 102, 108, 112] [30, 49, 53, 57, 78, 92, 97, 109] [24, 50, 52, 58, 98, 104, 110, 114] [25, 32, 45, 51, 80, 94, 99, 111] [6, 29, 52, 58, 112, 115, 118, 119] [7, 17, 30, 53, 90, 96, 113, 116] [31, 54, 55, 59, 91, 97, 114, 117] [0, 12, 55, 59, 90, 103, 107, 115] [11, 23, 36, 56, 108, 112, 116, 118] [37, 43, 53, 57, 92, 105, 109, 117] [16, 29, 42, 58, 102, 107, 118, 119] [4, 12, 47, 59, 103, 113, 115, 119]
Code ID 120-42-3 · download JSON · raw on GitHub