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[[126,18,7]] d ≤
n
126
k
18
d
7
kd²/n
7.0
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 7, d_Z ≤ 7 · 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 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[5, 30, 39, 67, 68, 104, 108]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[0, 16, 51, 58, 94, 98, 118]
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 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)×63 (2,4)×252 (3,3)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 63 (1,4): 63 (2,4): 252 (2,5): 756 (2,6): 252 (3,3): 63 (3,4): 315 (3,5): 1323 (3,6): 6300 (3,7): 8001 (3,8): 2142 (3,9): 756
trapping sets H_Z (1,3)×63 (2,4)×252 (3,3)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 63 (1,4): 63 (2,4): 252 (2,5): 756 (2,6): 252 (3,3): 63 (3,4): 378 (3,5): 1323 (3,6): 6048 (3,7): 8001 (3,8): 2457 (3,9): 756

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_wt7wtL3_order63_k18.txt, line 3. nonidentity supports a=[9, 34], b=[4, 5, 12].
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

[[126,18,7]] — two-block group-algebra code on SmallGroup(63,1)

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 = 7.00.

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_order63_k18.txt, line 3. This row is SmallGroup(63,1) with nonidentity GAP supports a=[9, 34], b=[4, 5, 12]. 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=18), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 7, not an exact certificate. The published distance (d=7) 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=[9, 34], b=[4, 5, 12] 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) = 18. 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 7) · Z-checks 63 (max weight 7)
H_X (63 checks, sparse supports)
[0, 34, 58, 63, 75, 107, 119] [1, 13, 56, 64, 81, 113, 122] [2, 42, 44, 64, 65, 104, 115] [3, 7, 43, 63, 66, 75, 93] [4, 28, 45, 63, 67, 98, 118] [5, 21, 41, 67, 68, 110, 120] [6, 22, 61, 64, 69, 81, 99] [7, 26, 52, 68, 70, 112, 121] [0, 8, 51, 64, 65, 71, 77] [9, 15, 52, 66, 72, 93, 111] [10, 29, 36, 65, 73, 106, 123] [11, 37, 53, 63, 67, 72, 74] [6, 12, 49, 73, 75, 117, 124] [13, 30, 50, 67, 68, 76, 83] [12, 14, 31, 69, 77, 99, 116] [2, 15, 35, 68, 70, 78, 85] [3, 16, 58, 71, 77, 79, 95] [2, 17, 24, 72, 80, 111, 124] [18, 20, 37, 70, 81, 114, 125] [19, 38, 45, 65, 73, 79, 82] [20, 46, 59, 72, 74, 83, 89] [14, 21, 57, 73, 75, 84, 91] [1, 22, 39, 76, 83, 85, 101] [21, 23, 40, 77, 86, 116, 125] [8, 24, 44, 78, 85, 87, 103] [7, 9, 25, 79, 88, 95, 113] [8, 26, 33, 80, 84, 89, 124] [27, 29, 46, 70, 81, 87, 90] [28, 47, 53, 79, 82, 91, 97] [29, 54, 62, 83, 89, 92, 107] [5, 23, 30, 84, 91, 93, 109] [6, 31, 48, 85, 94, 101, 118] [30, 32, 49, 86, 90, 95, 125] [0, 16, 33, 87, 96, 103, 120] [15, 17, 34, 69, 88, 97, 113] [16, 35, 42, 84, 89, 98, 102] [36, 38, 54, 87, 90, 99, 105] [37, 55, 59, 91, 97, 100, 115] [18, 38, 60, 66, 92, 101, 107] [13, 32, 39, 93, 102, 109, 123] [14, 40, 56, 74, 94, 103, 118] [39, 41, 57, 90, 95, 104, 108] [3, 25, 42, 76, 96, 105, 120] [24, 26, 43, 69, 86, 97, 106] [25, 44, 51, 98, 102, 107, 119] [45, 47, 60, 99, 105, 108, 121] [4, 46, 62, 71, 100, 109, 115] [10, 27, 47, 66, 80, 101, 110] [22, 41, 48, 82, 102, 111, 123] [23, 49, 61, 74, 92, 103, 112] [5, 48, 50, 104, 108, 113, 122] [9, 34, 51, 76, 94, 105, 114] [33, 35, 52, 86, 104, 106, 115] [10, 53, 55, 78, 108, 116, 121] [11, 18, 54, 71, 88, 109, 117] [19, 36, 55, 80, 98, 110, 118] [31, 50, 56, 82, 100, 111, 119] [12, 32, 57, 92, 110, 112, 120] [17, 43, 58, 94, 112, 114, 121] [4, 19, 59, 78, 96, 116, 122] [20, 27, 60, 88, 106, 117, 123] [1, 40, 61, 100, 117, 119, 124] [11, 28, 62, 96, 114, 122, 125]
H_Z (63 checks, sparse supports)
[0, 3, 4, 11, 63, 71, 96] [1, 2, 6, 8, 64, 85, 124] [2, 8, 10, 19, 65, 78, 80] [3, 9, 38, 47, 66, 79, 105] [4, 5, 11, 13, 67, 109, 122] [5, 7, 13, 15, 68, 93, 113] [6, 14, 34, 43, 69, 75, 94] [7, 15, 18, 27, 66, 70, 88] [8, 16, 46, 54, 71, 87, 89] [9, 11, 17, 20, 72, 88, 114] [10, 12, 19, 21, 73, 110, 116] [11, 20, 40, 49, 74, 117, 125] [0, 3, 12, 21, 75, 77, 120] [13, 22, 42, 51, 64, 76, 102] [8, 14, 16, 23, 77, 84, 103] [15, 24, 53, 59, 72, 78, 97] [16, 19, 25, 28, 79, 96, 98] [17, 26, 47, 55, 80, 97, 121] [1, 6, 18, 27, 81, 101, 117] [19, 28, 48, 56, 82, 118, 122] [13, 20, 22, 29, 81, 83, 123] [21, 26, 30, 35, 68, 84, 86] [15, 22, 24, 31, 69, 85, 111] [23, 32, 43, 52, 86, 93, 112] [24, 27, 33, 36, 80, 87, 106] [25, 34, 54, 60, 88, 105, 107] [20, 26, 29, 35, 70, 89, 106] [27, 32, 36, 41, 90, 110, 123] [21, 28, 30, 37, 67, 91, 125] [29, 38, 49, 57, 73, 90, 92] [3, 9, 30, 39, 76, 93, 95] [31, 40, 51, 58, 77, 94, 119] [16, 25, 32, 41, 95, 102, 120] [33, 42, 59, 62, 89, 96, 115] [28, 34, 37, 43, 63, 97, 114] [4, 35, 44, 55, 78, 98, 115] [6, 14, 36, 45, 73, 99, 118] [37, 46, 56, 61, 74, 81, 100] [22, 31, 38, 47, 82, 99, 101] [35, 39, 44, 48, 85, 102, 104] [24, 33, 40, 49, 86, 103, 124] [2, 41, 50, 52, 68, 104, 111] [36, 42, 45, 51, 65, 98, 105] [10, 43, 52, 60, 66, 106, 121] [0, 29, 38, 44, 65, 87, 107] [41, 45, 50, 53, 67, 82, 108] [30, 39, 46, 54, 83, 90, 109] [5, 47, 55, 57, 91, 108, 110] [9, 17, 48, 56, 94, 111, 113] [7, 49, 57, 58, 75, 95, 112] [1, 25, 34, 50, 76, 113, 119] [18, 51, 58, 62, 71, 107, 114] [2, 37, 46, 52, 70, 72, 115] [14, 23, 53, 59, 74, 91, 116] [12, 54, 60, 61, 92, 99, 117] [4, 31, 40, 55, 100, 116, 118] [0, 44, 56, 61, 64, 103, 119] [5, 33, 42, 57, 84, 104, 120] [7, 45, 53, 58, 63, 79, 121] [1, 50, 59, 62, 83, 100, 122] [10, 39, 48, 60, 101, 108, 123] [12, 17, 26, 61, 69, 112, 124] [18, 23, 32, 62, 92, 109, 125]
Code ID 126-18-7 · download JSON · raw on GitHub