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[[126,14,9]] d ≤
n
126
k
14
d
9
kd²/n
9.0
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 9, d_Z ≤ 9 · 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 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[9, 14, 16, 20, 22, 24, 28, 30, 36]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[67, 69, 73, 76, 81, 84, 98, 106, 114]
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 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)×189 (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): 189 (2,5): 756 (2,6): 378 (3,3): 63 (3,5): 945 (3,6): 6678 (3,7): 8757 (3,8): 3402 (3,9): 1134 (3,10): 252
trapping sets H_Z (1,3)×63 (2,4)×189 (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): 189 (2,5): 756 (2,6): 378 (3,3): 63 (3,5): 945 (3,6): 6678 (3,7): 8757 (3,8): 3402 (3,9): 1134 (3,10): 252

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction contributed via qldpc submit
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-18
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

[[126,14,9]] — two-block group-algebra code on SmallGroup(63,3)

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 = 9.00.

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_k14.txt, line 3. This row is SmallGroup(63,3) with nonidentity GAP supports a=[9, 34], b=[2, 6, 20]. 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=14), 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

Union Alpha (maker currently anonymous), 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(63,3), 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] and b=[2, 6, 20] 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) = 14. 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, 67, 81, 119] [1, 13, 56, 63, 64, 70, 122] [2, 42, 44, 65, 67, 73, 104] [3, 7, 43, 66, 75, 101, 116] [4, 28, 45, 64, 67, 75, 114] [5, 21, 41, 63, 65, 68, 110] [6, 22, 61, 69, 81, 89, 105] [7, 26, 52, 70, 73, 81, 112] [0, 8, 51, 64, 71, 101, 109] [9, 15, 52, 72, 74, 90, 93] [10, 29, 36, 64, 68, 73, 98] [11, 37, 53, 70, 74, 95, 111] [6, 12, 49, 65, 70, 75, 117] [13, 30, 50, 67, 76, 89, 97] [12, 14, 31, 66, 77, 78, 99] [2, 15, 35, 68, 78, 109, 116] [3, 16, 58, 74, 77, 79, 82] [2, 17, 24, 80, 110, 111, 122] [18, 20, 37, 68, 75, 81, 106] [19, 38, 45, 63, 82, 95, 103] [20, 46, 59, 69, 83, 84, 87] [14, 21, 57, 73, 84, 97, 105] [1, 22, 39, 66, 71, 83, 85] [21, 23, 40, 86, 98, 114, 116] [8, 24, 44, 82, 85, 87, 90] [7, 9, 25, 88, 95, 110, 117] [8, 26, 33, 83, 89, 99, 124] [27, 29, 46, 65, 90, 103, 111] [28, 47, 53, 69, 72, 76, 91] [29, 54, 62, 92, 104, 105, 119] [5, 23, 30, 71, 78, 91, 93] [6, 31, 48, 94, 98, 101, 106] [30, 32, 49, 72, 87, 95, 125] [0, 16, 33, 96, 103, 117, 122] [15, 17, 34, 83, 91, 97, 113] [16, 35, 42, 84, 98, 118, 125] [36, 38, 54, 76, 79, 84, 99] [37, 55, 59, 89, 100, 104, 112] [18, 38, 60, 77, 93, 101, 121] [13, 32, 39, 102, 106, 109, 114] [14, 40, 56, 72, 79, 103, 118] [39, 41, 57, 90, 104, 107, 121] [3, 25, 42, 91, 99, 105, 120] [24, 26, 43, 69, 106, 118, 123] [25, 44, 51, 92, 102, 107, 108] [45, 47, 60, 97, 108, 112, 119] [4, 46, 62, 77, 85, 107, 109] [10, 27, 47, 78, 110, 113, 124] [22, 41, 48, 79, 87, 111, 123] [23, 49, 61, 74, 107, 112, 115] [5, 48, 50, 80, 96, 108, 113] [9, 34, 51, 76, 114, 123, 125] [33, 35, 52, 86, 92, 100, 115] [10, 53, 55, 85, 93, 115, 116] [11, 18, 54, 66, 113, 117, 120] [19, 36, 55, 86, 96, 102, 118] [31, 50, 56, 82, 115, 119, 121] [12, 32, 57, 80, 88, 92, 120] [17, 43, 58, 94, 100, 108, 121] [4, 19, 59, 71, 120, 122, 124] [20, 27, 60, 80, 86, 94, 123] [1, 40, 61, 88, 96, 100, 124] [11, 28, 62, 88, 94, 102, 125]
H_Z (63 checks, sparse supports)
[0, 1, 5, 19, 63, 71, 96] [1, 4, 8, 10, 64, 85, 124] [2, 5, 12, 27, 65, 78, 80] [3, 14, 22, 54, 66, 79, 105] [0, 2, 4, 13, 67, 109, 122] [5, 10, 15, 18, 68, 93, 113] [6, 20, 28, 43, 69, 75, 94] [1, 7, 11, 12, 66, 70, 88] [8, 22, 30, 59, 71, 87, 89] [9, 28, 32, 40, 72, 88, 114] [2, 7, 10, 21, 73, 110, 116] [9, 11, 16, 49, 74, 117, 125] [3, 4, 12, 18, 75, 77, 120] [13, 28, 36, 51, 64, 76, 102] [14, 16, 38, 46, 77, 84, 103] [14, 15, 30, 47, 72, 78, 97] [16, 36, 40, 48, 79, 96, 98] [17, 50, 57, 60, 80, 97, 121] [0, 6, 7, 18, 81, 101, 117] [16, 19, 24, 56, 82, 118, 122] [20, 22, 26, 34, 81, 83, 123] [20, 21, 35, 36, 68, 84, 86] [22, 24, 46, 53, 69, 85, 111] [23, 52, 55, 60, 86, 93, 112] [20, 24, 32, 48, 80, 87, 106] [25, 57, 61, 62, 88, 105, 107] [6, 13, 26, 37, 70, 89, 106] [9, 24, 27, 41, 90, 110, 123] [28, 30, 34, 42, 67, 91, 125] [29, 44, 52, 57, 73, 90, 92] [9, 30, 38, 53, 76, 93, 95] [31, 58, 60, 62, 77, 94, 119] [11, 19, 25, 32, 95, 102, 120] [33, 50, 55, 61, 89, 96, 115] [13, 21, 34, 45, 63, 97, 114] [10, 23, 31, 35, 78, 98, 115] [14, 26, 36, 42, 73, 99, 118] [37, 52, 58, 61, 74, 81, 100] [3, 8, 31, 38, 82, 99, 101] [39, 44, 55, 62, 85, 102, 104] [19, 27, 33, 40, 86, 103, 124] [2, 29, 37, 41, 68, 104, 111] [6, 21, 29, 42, 65, 98, 105] [18, 31, 39, 43, 66, 106, 121] [41, 44, 46, 49, 65, 87, 107] [44, 45, 50, 58, 67, 82, 108] [8, 15, 39, 46, 83, 90, 109] [5, 17, 25, 47, 91, 108, 110] [11, 17, 27, 48, 94, 111, 113] [7, 37, 45, 49, 75, 95, 112] [34, 47, 50, 54, 76, 113, 119] [4, 23, 39, 51, 71, 107, 114] [49, 52, 53, 56, 70, 72, 115] [3, 15, 23, 53, 74, 91, 116] [12, 25, 33, 54, 92, 99, 117] [35, 40, 43, 55, 100, 116, 118] [0, 29, 45, 56, 64, 103, 119] [42, 54, 57, 59, 84, 104, 120] [38, 41, 56, 58, 63, 79, 121] [1, 17, 33, 59, 83, 100, 122] [43, 48, 51, 60, 101, 108, 123] [26, 47, 59, 61, 69, 112, 124] [32, 35, 51, 62, 92, 109, 125]
Code ID 126-14-9 · download JSON · raw on GitHub