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[[126,6,9]] d ≤
n
126
k
6
d
9
kd²/n
3.857
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 9, d_Z ≤ 9 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[89, 95, 97, 101, 103, 105, 109, 111, 116]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[0, 1, 2, 4, 5, 7, 10, 12, 18]
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)
trapping sets H_X (1,2)×63 (2,2)×63 (3,2)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 63 (1,3): 63 (2,2): 63 (2,3): 378 (2,4): 189 (3,2): 63 (3,3): 1197 (3,4): 2079 (3,5): 756 (3,6): 378 (3,7): 63
trapping sets H_Z (1,2)×63 (2,2)×63 (3,2)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 63 (1,3): 63 (2,2): 63 (2,3): 378 (2,4): 189 (3,2): 63 (3,3): 1197 (3,4): 2079 (3,5): 756 (3,6): 378 (3,7): 63

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(63,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt5wtL2_order63_k6.txt, line 3. nonidentity supports a=[2], b=[9, 34].
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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[126,6,9]] — two-block group-algebra code on SmallGroup(63,1)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-5 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 3.86.

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_wt5wtL2_order63_k6.txt, line 3. This row is SmallGroup(63,1) with nonidentity GAP supports a=[2], b=[9, 34]. 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=6), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 9, not an exact certificate. The published distance (d=9) 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(63,1), 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], b=[9, 34] 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 = 126 - rank(HX) - rank(HZ) = 6. 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 63 (max weight 5) · Z-checks 63 (max weight 5)
H_X (63 checks, sparse supports)
[0, 18, 63, 97, 121] [0, 1, 64, 103, 124] [2, 4, 65, 105, 107] [3, 27, 66, 70, 106] [1, 4, 67, 109, 125] [2, 5, 68, 111, 113] [3, 6, 69, 75, 112] [7, 10, 70, 89, 115] [8, 11, 63, 71, 114] [9, 36, 72, 78, 115] [5, 10, 73, 116, 118] [6, 11, 74, 81, 117] [7, 12, 75, 95, 120] [8, 13, 64, 76, 119] [9, 14, 77, 84, 120] [15, 19, 65, 78, 98] [16, 20, 66, 79, 121] [17, 45, 65, 80, 87] [12, 18, 81, 101, 123] [13, 19, 67, 82, 122] [14, 20, 83, 90, 123] [15, 21, 68, 84, 104] [16, 22, 69, 85, 124] [17, 23, 68, 86, 93] [24, 28, 71, 87, 107] [25, 29, 70, 72, 88] [26, 53, 71, 89, 96] [21, 27, 73, 90, 110] [22, 28, 74, 91, 125] [23, 29, 73, 92, 99] [24, 30, 76, 93, 113] [25, 31, 75, 77, 94] [26, 32, 76, 95, 102] [33, 37, 63, 79, 96] [34, 38, 78, 80, 97] [35, 59, 79, 98, 105] [30, 36, 82, 99, 118] [31, 37, 81, 83, 100] [32, 38, 82, 101, 108] [33, 39, 64, 85, 102] [34, 40, 84, 86, 103] [35, 41, 85, 104, 111] [42, 46, 66, 88, 105] [43, 47, 87, 89, 106] [44, 62, 88, 107, 114] [39, 45, 67, 91, 108] [40, 46, 90, 92, 109] [41, 47, 91, 110, 116] [42, 48, 69, 94, 111] [43, 49, 93, 95, 112] [44, 50, 94, 113, 119] [51, 54, 72, 97, 114] [52, 55, 96, 98, 115] [48, 53, 74, 100, 116] [49, 54, 99, 101, 117] [50, 55, 100, 118, 122] [51, 56, 77, 103, 119] [52, 57, 102, 104, 120] [58, 60, 80, 106, 121] [56, 59, 83, 109, 122] [57, 60, 108, 110, 123] [58, 61, 86, 112, 124] [61, 62, 92, 117, 125]
H_Z (63 checks, sparse supports)
[0, 8, 33, 63, 64] [1, 13, 39, 64, 67] [2, 15, 17, 65, 68] [3, 16, 42, 66, 69] [4, 19, 45, 65, 67] [5, 21, 23, 68, 73] [6, 22, 48, 69, 74] [3, 7, 25, 70, 75] [8, 24, 26, 71, 76] [9, 25, 51, 72, 77] [10, 27, 29, 70, 73] [11, 28, 53, 71, 74] [6, 12, 31, 75, 81] [13, 30, 32, 76, 82] [14, 31, 56, 77, 83] [9, 15, 34, 78, 84] [16, 33, 35, 79, 85] [17, 34, 58, 80, 86] [11, 18, 37, 63, 81] [19, 36, 38, 78, 82] [20, 37, 59, 79, 83] [14, 21, 40, 84, 90] [22, 39, 41, 85, 91] [23, 40, 61, 86, 92] [17, 24, 43, 87, 93] [25, 42, 44, 88, 94] [7, 26, 43, 89, 95] [20, 27, 46, 66, 90] [28, 45, 47, 87, 91] [29, 46, 62, 88, 92] [23, 30, 49, 93, 99] [31, 48, 50, 94, 100] [12, 32, 49, 95, 101] [26, 33, 52, 96, 102] [0, 34, 51, 97, 103] [15, 35, 52, 98, 104] [29, 36, 54, 72, 99] [37, 53, 55, 96, 100] [18, 38, 54, 97, 101] [32, 39, 57, 102, 108] [1, 40, 56, 103, 109] [21, 41, 57, 104, 110] [2, 35, 42, 105, 111] [3, 43, 58, 106, 112] [2, 24, 44, 107, 113] [38, 45, 60, 80, 108] [4, 46, 59, 105, 109] [27, 47, 60, 106, 110] [5, 41, 48, 111, 116] [6, 49, 61, 112, 117] [5, 30, 50, 113, 118] [8, 44, 51, 114, 119] [7, 9, 52, 115, 120] [10, 47, 53, 89, 116] [11, 54, 62, 114, 117] [10, 36, 55, 115, 118] [13, 50, 56, 119, 122] [12, 14, 57, 120, 123] [0, 16, 58, 121, 124] [19, 55, 59, 98, 122] [18, 20, 60, 121, 123] [1, 22, 61, 124, 125] [4, 28, 62, 107, 125]
Code ID 126-6-9 · download JSON · raw on GitHub