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[[124,4,16]] d ≤
n
124
k
4
d
16
kd²/n
8.258
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[6, 12, 18, 21, 25, 27, 36, 37, 45, 63, 77, 78, 82, 83, 97, 116]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[2, 3, 7, 9, 12, 13, 15, 19, 42, 62, 71, 79, 98, 116, 118, 123]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

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)×124 (2,4)×186 (3,4)×124 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 124 (2,4): 186 (2,6): 1364 (3,4): 124 (3,6): 5270 (3,8): 21266 (3,10): 1736
trapping sets H_Z (1,4)×124 (2,4)×186 (3,4)×124 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 124 (2,4): 186 (2,6): 1364 (3,4): 124 (3,6): 5270 (3,8): 21266 (3,10): 1736

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(62,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order62_k4.txt, line 10. nonidentity supports a=[2, 3, 12], b=[3, 7, 45].
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

[[124,4,16]] — two-block group-algebra code on SmallGroup(62,1)

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 = 8.26.

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_order62_k4.txt, line 10. This row is SmallGroup(62,1) with nonidentity GAP supports a=[2, 3, 12], b=[3, 7, 45]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 16, 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(62,1), 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, 3, 12], b=[3, 7, 45] 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 = 124 - rank(HX) - rank(HZ) = 4. 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 62 (max weight 8) · Z-checks 62 (max weight 8)
H_X (62 checks, sparse supports)
[0, 1, 11, 60, 62, 80, 118, 122] [0, 1, 3, 52, 63, 81, 119, 123] [0, 2, 3, 13, 62, 64, 82, 120] [2, 3, 5, 54, 63, 65, 83, 121] [2, 4, 5, 15, 64, 66, 84, 122] [4, 5, 7, 56, 65, 67, 85, 123] [4, 6, 7, 17, 62, 66, 68, 86] [6, 7, 9, 58, 63, 67, 69, 87] [6, 8, 9, 19, 64, 68, 70, 88] [8, 9, 11, 60, 65, 69, 71, 89] [8, 10, 11, 21, 66, 70, 72, 90] [0, 10, 11, 13, 67, 71, 73, 91] [10, 12, 13, 23, 68, 72, 74, 92] [2, 12, 13, 15, 69, 73, 75, 93] [12, 14, 15, 25, 70, 74, 76, 94] [4, 14, 15, 17, 71, 75, 77, 95] [14, 16, 17, 27, 72, 76, 78, 96] [6, 16, 17, 19, 73, 77, 79, 97] [16, 18, 19, 29, 74, 78, 80, 98] [8, 18, 19, 21, 75, 79, 81, 99] [18, 20, 21, 31, 76, 80, 82, 100] [10, 20, 21, 23, 77, 81, 83, 101] [20, 22, 23, 33, 78, 82, 84, 102] [12, 22, 23, 25, 79, 83, 85, 103] [22, 24, 25, 35, 80, 84, 86, 104] [14, 24, 25, 27, 81, 85, 87, 105] [24, 26, 27, 37, 82, 86, 88, 106] [16, 26, 27, 29, 83, 87, 89, 107] [26, 28, 29, 39, 84, 88, 90, 108] [18, 28, 29, 31, 85, 89, 91, 109] [28, 30, 31, 41, 86, 90, 92, 110] [20, 30, 31, 33, 87, 91, 93, 111] [30, 32, 33, 43, 88, 92, 94, 112] [22, 32, 33, 35, 89, 93, 95, 113] [32, 34, 35, 45, 90, 94, 96, 114] [24, 34, 35, 37, 91, 95, 97, 115] [34, 36, 37, 47, 92, 96, 98, 116] [26, 36, 37, 39, 93, 97, 99, 117] [36, 38, 39, 49, 94, 98, 100, 118] [28, 38, 39, 41, 95, 99, 101, 119] [38, 40, 41, 51, 96, 100, 102, 120] [30, 40, 41, 43, 97, 101, 103, 121] [40, 42, 43, 53, 98, 102, 104, 122] [32, 42, 43, 45, 99, 103, 105, 123] [42, 44, 45, 55, 62, 100, 104, 106] [34, 44, 45, 47, 63, 101, 105, 107] [44, 46, 47, 57, 64, 102, 106, 108] [36, 46, 47, 49, 65, 103, 107, 109] [46, 48, 49, 59, 66, 104, 108, 110] [38, 48, 49, 51, 67, 105, 109, 111] [48, 50, 51, 61, 68, 106, 110, 112] [40, 50, 51, 53, 69, 107, 111, 113] [1, 50, 52, 53, 70, 108, 112, 114] [42, 52, 53, 55, 71, 109, 113, 115] [3, 52, 54, 55, 72, 110, 114, 116] [44, 54, 55, 57, 73, 111, 115, 117] [5, 54, 56, 57, 74, 112, 116, 118] [46, 56, 57, 59, 75, 113, 117, 119] [7, 56, 58, 59, 76, 114, 118, 120] [48, 58, 59, 61, 77, 115, 119, 121] [9, 58, 60, 61, 78, 116, 120, 122] [1, 50, 60, 61, 79, 117, 121, 123]
H_Z (62 checks, sparse supports)
[0, 2, 6, 44, 62, 63, 64, 73] [1, 3, 7, 45, 62, 63, 114, 123] [2, 4, 8, 46, 64, 65, 66, 75] [3, 5, 9, 47, 63, 64, 65, 116] [4, 6, 10, 48, 66, 67, 68, 77] [5, 7, 11, 49, 65, 66, 67, 118] [6, 8, 12, 50, 68, 69, 70, 79] [7, 9, 13, 51, 67, 68, 69, 120] [8, 10, 14, 52, 70, 71, 72, 81] [9, 11, 15, 53, 69, 70, 71, 122] [10, 12, 16, 54, 72, 73, 74, 83] [11, 13, 17, 55, 62, 71, 72, 73] [12, 14, 18, 56, 74, 75, 76, 85] [13, 15, 19, 57, 64, 73, 74, 75] [14, 16, 20, 58, 76, 77, 78, 87] [15, 17, 21, 59, 66, 75, 76, 77] [16, 18, 22, 60, 78, 79, 80, 89] [17, 19, 23, 61, 68, 77, 78, 79] [0, 18, 20, 24, 80, 81, 82, 91] [1, 19, 21, 25, 70, 79, 80, 81] [2, 20, 22, 26, 82, 83, 84, 93] [3, 21, 23, 27, 72, 81, 82, 83] [4, 22, 24, 28, 84, 85, 86, 95] [5, 23, 25, 29, 74, 83, 84, 85] [6, 24, 26, 30, 86, 87, 88, 97] [7, 25, 27, 31, 76, 85, 86, 87] [8, 26, 28, 32, 88, 89, 90, 99] [9, 27, 29, 33, 78, 87, 88, 89] [10, 28, 30, 34, 90, 91, 92, 101] [11, 29, 31, 35, 80, 89, 90, 91] [12, 30, 32, 36, 92, 93, 94, 103] [13, 31, 33, 37, 82, 91, 92, 93] [14, 32, 34, 38, 94, 95, 96, 105] [15, 33, 35, 39, 84, 93, 94, 95] [16, 34, 36, 40, 96, 97, 98, 107] [17, 35, 37, 41, 86, 95, 96, 97] [18, 36, 38, 42, 98, 99, 100, 109] [19, 37, 39, 43, 88, 97, 98, 99] [20, 38, 40, 44, 100, 101, 102, 111] [21, 39, 41, 45, 90, 99, 100, 101] [22, 40, 42, 46, 102, 103, 104, 113] [23, 41, 43, 47, 92, 101, 102, 103] [24, 42, 44, 48, 104, 105, 106, 115] [25, 43, 45, 49, 94, 103, 104, 105] [26, 44, 46, 50, 106, 107, 108, 117] [27, 45, 47, 51, 96, 105, 106, 107] [28, 46, 48, 52, 108, 109, 110, 119] [29, 47, 49, 53, 98, 107, 108, 109] [30, 48, 50, 54, 110, 111, 112, 121] [31, 49, 51, 55, 100, 109, 110, 111] [32, 50, 52, 56, 112, 113, 114, 123] [33, 51, 53, 57, 102, 111, 112, 113] [34, 52, 54, 58, 63, 114, 115, 116] [35, 53, 55, 59, 104, 113, 114, 115] [36, 54, 56, 60, 65, 116, 117, 118] [37, 55, 57, 61, 106, 115, 116, 117] [0, 38, 56, 58, 67, 118, 119, 120] [1, 39, 57, 59, 108, 117, 118, 119] [2, 40, 58, 60, 69, 120, 121, 122] [3, 41, 59, 61, 110, 119, 120, 121] [0, 4, 42, 60, 62, 71, 122, 123] [1, 5, 43, 61, 112, 121, 122, 123]
Code ID 124-4-16 · download JSON · raw on GitHub