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[[120,36,2]] d =
n
120
k
36
d
2
kd²/n
1.2
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, 70]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[17, 27]
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)×60 (3,3)×20 (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): 60 (2,4): 60 (2,5): 720 (2,6): 240 (3,3): 20 (3,4): 840 (3,5): 720 (3,6): 3780 (3,7): 7560 (3,8): 1020 (3,9): 720 (3,10): 60
trapping sets H_Z (1,3)×60 (2,0)×60 (3,3)×20 (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): 60 (2,4): 60 (2,5): 720 (2,6): 240 (3,3): 20 (3,4): 840 (3,5): 720 (3,6): 3780 (3,7): 7560 (3,8): 1020 (3,9): 720 (3,10): 60

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(60,6), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt7wtL3_order60_k36.txt, line 1. nonidentity supports a=[3, 9], b=[2, 5, 47].
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

[[120,36,2]] — two-block group-algebra code on SmallGroup(60,6)

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 = 1.20.

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_k36.txt, line 1. This row is SmallGroup(60,6) with nonidentity GAP supports a=[3, 9], b=[2, 5, 47]. 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=36), 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(60,6), 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=[3, 9], b=[2, 5, 47] 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) = 36. 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, 2, 8, 60, 66, 95, 106] [1, 5, 13, 60, 61, 101, 111] [0, 2, 8, 62, 74, 105, 114] [3, 9, 18, 61, 63, 95, 106] [4, 10, 19, 60, 64, 94, 100] [1, 5, 13, 62, 65, 110, 117] [6, 14, 24, 63, 66, 101, 111] [7, 15, 25, 61, 67, 83, 100] [0, 2, 8, 68, 84, 113, 118] [3, 9, 18, 65, 69, 105, 114] [4, 10, 19, 62, 70, 104, 109] [11, 20, 30, 63, 71, 83, 89] [12, 21, 31, 64, 72, 76, 82] [1, 5, 13, 68, 73, 116, 119] [6, 14, 24, 69, 74, 110, 117] [7, 15, 25, 65, 75, 93, 109] [16, 26, 36, 66, 76, 89, 94] [17, 27, 37, 64, 67, 77, 88] [3, 9, 18, 73, 78, 113, 118] [4, 10, 19, 68, 79, 112, 115] [11, 20, 30, 69, 80, 93, 99] [12, 21, 31, 70, 81, 86, 92] [22, 32, 42, 67, 71, 72, 82] [23, 33, 43, 71, 72, 83, 111] [6, 14, 24, 78, 84, 116, 119] [7, 15, 25, 73, 85, 103, 115] [16, 26, 36, 74, 86, 99, 104] [17, 27, 37, 70, 75, 87, 98] [28, 38, 47, 71, 76, 77, 88] [29, 39, 48, 76, 77, 89, 95] [11, 20, 30, 78, 90, 103, 108] [12, 21, 31, 79, 91, 96, 102] [22, 32, 42, 75, 80, 81, 92] [23, 33, 43, 80, 81, 93, 117] [34, 44, 52, 64, 82, 94, 101] [35, 45, 53, 63, 83, 88, 95] [16, 26, 36, 84, 96, 108, 112] [17, 27, 37, 79, 85, 97, 107] [28, 38, 47, 80, 86, 87, 98] [29, 39, 48, 86, 87, 99, 105] [40, 49, 55, 67, 88, 100, 106] [41, 50, 56, 66, 72, 89, 101] [22, 32, 42, 85, 90, 91, 102] [23, 33, 43, 90, 91, 103, 119] [34, 44, 52, 70, 92, 104, 110] [35, 45, 53, 69, 93, 98, 105] [46, 54, 58, 60, 77, 94, 106] [28, 38, 47, 90, 96, 97, 107] [29, 39, 48, 96, 97, 108, 113] [40, 49, 55, 75, 98, 109, 114] [41, 50, 56, 74, 81, 99, 110] [51, 57, 59, 61, 82, 100, 111] [34, 44, 52, 79, 102, 112, 116] [35, 45, 53, 78, 103, 107, 113] [46, 54, 58, 62, 87, 104, 114] [40, 49, 55, 85, 107, 115, 118] [41, 50, 56, 84, 91, 108, 116] [51, 57, 59, 65, 92, 109, 117] [46, 54, 58, 68, 97, 112, 118] [51, 57, 59, 73, 102, 115, 119]
H_Z (60 checks, sparse supports)
[0, 1, 4, 46, 60, 62, 68] [1, 3, 7, 51, 61, 65, 73] [2, 5, 10, 54, 60, 62, 68] [3, 6, 11, 35, 63, 69, 78] [4, 12, 17, 34, 64, 70, 79] [5, 9, 15, 57, 61, 65, 73] [0, 6, 16, 41, 66, 74, 84] [7, 17, 22, 40, 67, 75, 85] [8, 13, 19, 58, 60, 62, 68] [9, 14, 20, 45, 63, 69, 78] [10, 21, 27, 44, 64, 70, 79] [11, 22, 23, 28, 71, 80, 90] [12, 22, 23, 41, 72, 81, 91] [13, 18, 25, 59, 61, 65, 73] [2, 14, 26, 50, 66, 74, 84] [15, 27, 32, 49, 67, 75, 85] [12, 16, 28, 29, 76, 86, 96] [17, 28, 29, 46, 77, 87, 97] [18, 24, 30, 53, 63, 69, 78] [19, 31, 37, 52, 64, 70, 79] [20, 32, 33, 38, 71, 80, 90] [21, 32, 33, 50, 72, 81, 91] [12, 22, 34, 51, 82, 92, 102] [7, 11, 23, 35, 83, 93, 103] [8, 24, 36, 56, 66, 74, 84] [25, 37, 42, 55, 67, 75, 85] [21, 26, 38, 39, 76, 86, 96] [27, 38, 39, 54, 77, 87, 97] [17, 28, 35, 40, 88, 98, 107] [11, 16, 29, 41, 89, 99, 108] [30, 42, 43, 47, 71, 80, 90] [31, 42, 43, 56, 72, 81, 91] [21, 32, 44, 57, 82, 92, 102] [15, 20, 33, 45, 83, 93, 103] [4, 16, 34, 46, 94, 104, 112] [0, 3, 29, 35, 95, 105, 113] [31, 36, 47, 48, 76, 86, 96] [37, 47, 48, 58, 77, 87, 97] [27, 38, 45, 49, 88, 98, 107] [20, 26, 39, 50, 89, 99, 108] [4, 7, 40, 51, 100, 109, 115] [1, 6, 34, 41, 101, 110, 116] [31, 42, 52, 59, 82, 92, 102] [25, 30, 43, 53, 83, 93, 103] [10, 26, 44, 54, 94, 104, 112] [2, 9, 39, 45, 95, 105, 113] [0, 3, 40, 46, 106, 114, 118] [37, 47, 53, 55, 88, 98, 107] [30, 36, 48, 56, 89, 99, 108] [10, 15, 49, 57, 100, 109, 115] [5, 14, 44, 50, 101, 110, 116] [1, 6, 23, 51, 111, 117, 119] [19, 36, 52, 58, 94, 104, 112] [8, 18, 48, 53, 95, 105, 113] [2, 9, 49, 54, 106, 114, 118] [19, 25, 55, 59, 100, 109, 115] [13, 24, 52, 56, 101, 110, 116] [5, 14, 33, 57, 111, 117, 119] [8, 18, 55, 58, 106, 114, 118] [13, 24, 43, 59, 111, 117, 119]
Code ID 120-36-2 · download JSON · raw on GitHub