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[[120,44,2]] d =
n
120
k
44
d
2
kd²/n
1.467
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)
[40, 51]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[7, 17]
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 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)
trapping sets H_X (1,3)×60 (2,0)×90 (3,3)×380 (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 (2,0): 90 (2,4): 120 (2,5): 720 (3,3): 380 (3,4): 1080 (3,5): 1440 (3,6): 3240 (3,7): 3600
trapping sets H_Z (1,3)×60 (2,0)×90 (3,3)×380 (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 (2,0): 90 (2,4): 120 (2,5): 720 (3,3): 380 (3,4): 1080 (3,5): 1440 (3,6): 3240 (3,7): 3600

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_wt7wtL3_order60_k44.txt, line 1. nonidentity supports a=[5, 13], b=[4, 6, 15].
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

[[120,44,2]] — 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 = 1.47.

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_k44.txt, line 1. This row is SmallGroup(60,1) with nonidentity GAP supports a=[5, 13], b=[4, 6, 15]. 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=44), 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=[5, 13], b=[4, 6, 15] 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) = 44. 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, 4, 12, 60, 63, 96, 107] [1, 7, 17, 61, 66, 90, 102] [2, 10, 22, 61, 62, 66, 69] [3, 11, 23, 60, 63, 96, 107] [0, 4, 12, 64, 71, 116, 119] [5, 15, 28, 60, 63, 65, 74] [6, 16, 29, 61, 66, 90, 102] [1, 7, 17, 67, 76, 113, 118] [8, 20, 34, 65, 68, 74, 79] [9, 21, 35, 61, 62, 66, 69] [2, 10, 22, 70, 77, 81, 89] [3, 11, 23, 64, 71, 116, 119] [0, 4, 12, 72, 83, 108, 115] [13, 26, 40, 62, 69, 73, 85] [14, 27, 41, 60, 63, 65, 74] [5, 15, 28, 72, 75, 83, 87] [6, 16, 29, 67, 76, 113, 118] [1, 7, 17, 77, 89, 103, 112] [18, 32, 45, 73, 78, 85, 91] [19, 33, 46, 65, 68, 74, 79] [8, 20, 34, 80, 88, 93, 101] [9, 21, 35, 70, 77, 81, 89] [2, 10, 22, 67, 76, 82, 95] [3, 11, 23, 72, 83, 108, 115] [24, 38, 50, 68, 79, 84, 97] [25, 39, 51, 62, 69, 73, 85] [13, 26, 40, 82, 86, 95, 99] [14, 27, 41, 72, 75, 83, 87] [5, 15, 28, 64, 71, 88, 101] [6, 16, 29, 77, 89, 103, 112] [30, 43, 53, 84, 90, 97, 102] [31, 44, 54, 73, 78, 85, 91] [18, 32, 45, 92, 100, 104, 111] [19, 33, 46, 80, 88, 93, 101] [8, 20, 34, 75, 87, 94, 106] [9, 21, 35, 67, 76, 82, 95] [36, 48, 56, 78, 91, 96, 107] [37, 49, 57, 68, 79, 84, 97] [24, 38, 50, 94, 98, 106, 109] [25, 39, 51, 82, 86, 95, 99] [13, 26, 40, 70, 81, 100, 111] [14, 27, 41, 64, 71, 88, 101] [42, 52, 58, 84, 90, 97, 102] [30, 43, 53, 103, 110, 112, 117] [31, 44, 54, 92, 100, 104, 111] [18, 32, 45, 86, 99, 105, 114] [19, 33, 46, 75, 87, 94, 106] [47, 55, 59, 78, 91, 96, 107] [36, 48, 56, 105, 108, 114, 115] [37, 49, 57, 94, 98, 106, 109] [24, 38, 50, 80, 93, 110, 117] [25, 39, 51, 70, 81, 100, 111] [42, 52, 58, 103, 110, 112, 117] [30, 43, 53, 98, 109, 113, 118] [31, 44, 54, 86, 99, 105, 114] [47, 55, 59, 105, 108, 114, 115] [36, 48, 56, 92, 104, 116, 119] [37, 49, 57, 80, 93, 110, 117] [42, 52, 58, 98, 109, 113, 118] [47, 55, 59, 92, 104, 116, 119]
H_Z (60 checks, sparse supports)
[0, 3, 5, 14, 60, 64, 72] [1, 2, 6, 9, 61, 67, 77] [2, 9, 13, 25, 62, 70, 82] [0, 3, 5, 14, 63, 71, 83] [4, 11, 28, 41, 60, 64, 72] [5, 8, 14, 19, 65, 75, 88] [1, 2, 6, 9, 66, 76, 89] [7, 16, 22, 35, 61, 67, 77] [8, 19, 24, 37, 68, 80, 94] [2, 9, 13, 25, 69, 81, 95] [10, 21, 40, 51, 62, 70, 82] [4, 11, 28, 41, 63, 71, 83] [12, 15, 23, 27, 60, 64, 72] [13, 18, 25, 31, 73, 86, 100] [5, 8, 14, 19, 74, 87, 101] [15, 27, 34, 46, 65, 75, 88] [7, 16, 22, 35, 66, 76, 89] [10, 17, 21, 29, 61, 67, 77] [18, 31, 36, 47, 78, 92, 105] [8, 19, 24, 37, 79, 93, 106] [20, 33, 50, 57, 68, 80, 94] [10, 21, 40, 51, 69, 81, 95] [22, 26, 35, 39, 62, 70, 82] [12, 15, 23, 27, 63, 71, 83] [24, 30, 37, 42, 84, 98, 110] [13, 18, 25, 31, 85, 99, 111] [26, 39, 45, 54, 73, 86, 100] [15, 27, 34, 46, 74, 87, 101] [20, 28, 33, 41, 65, 75, 88] [10, 17, 21, 29, 66, 76, 89] [1, 6, 30, 42, 90, 103, 113] [18, 31, 36, 47, 91, 104, 114] [32, 44, 56, 59, 78, 92, 105] [20, 33, 50, 57, 79, 93, 106] [34, 38, 46, 49, 68, 80, 94] [22, 26, 35, 39, 69, 81, 95] [0, 3, 36, 47, 96, 108, 116] [24, 30, 37, 42, 97, 109, 117] [38, 49, 53, 58, 84, 98, 110] [26, 39, 45, 54, 85, 99, 111] [32, 40, 44, 51, 73, 86, 100] [20, 28, 33, 41, 74, 87, 101] [1, 6, 30, 42, 102, 112, 118] [17, 29, 43, 52, 90, 103, 113] [32, 44, 56, 59, 91, 104, 114] [45, 48, 54, 55, 78, 92, 105] [34, 38, 46, 49, 79, 93, 106] [0, 3, 36, 47, 107, 115, 119] [12, 23, 48, 55, 96, 108, 116] [38, 49, 53, 58, 97, 109, 117] [43, 50, 52, 57, 84, 98, 110] [32, 40, 44, 51, 85, 99, 111] [17, 29, 43, 52, 102, 112, 118] [7, 16, 53, 58, 90, 103, 113] [45, 48, 54, 55, 91, 104, 114] [12, 23, 48, 55, 107, 115, 119] [4, 11, 56, 59, 96, 108, 116] [43, 50, 52, 57, 97, 109, 117] [7, 16, 53, 58, 102, 112, 118] [4, 11, 56, 59, 107, 115, 119]
Code ID 120-44-2 · download JSON · raw on GitHub