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[[126,40,2]] d =
n
126
k
40
d
2
kd²/n
1.27
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)
[80, 92]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[34, 46]
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)×63 (2,0)×63 (3,0)×21 (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 (2,0): 63 (2,4): 63 (2,5): 756 (2,6): 252 (3,0): 21 (3,3): 21 (3,4): 756 (3,5): 756 (3,6): 4158 (3,7): 7938 (3,8): 1008 (3,9): 756
trapping sets H_Z (1,3)×63 (2,0)×63 (3,0)×21 (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 (2,0): 63 (2,4): 63 (2,5): 756 (2,6): 252 (3,0): 21 (3,3): 21 (3,4): 756 (3,5): 756 (3,6): 4158 (3,7): 7938 (3,8): 1008 (3,9): 756

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(63,3), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt7wtL3_order63_k40.txt, line 1. nonidentity supports a=[2, 5], b=[4, 6, 28].
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

[[126,40,2]] — 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 = 1.27.

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_k40.txt, line 1. This row is SmallGroup(63,3) with nonidentity GAP supports a=[2, 5], b=[4, 6, 28]. 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=40), 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(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=[2, 5], b=[4, 6, 28] 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) = 40. 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, 1, 4, 63, 81, 107, 112] [0, 1, 4, 64, 70, 113, 117] [2, 5, 10, 65, 67, 115, 119] [3, 6, 11, 63, 66, 68, 116] [0, 1, 4, 67, 75, 106, 118] [2, 5, 10, 63, 68, 120, 122] [3, 6, 11, 64, 69, 73, 105] [7, 12, 18, 70, 73, 104, 121] [8, 13, 19, 65, 71, 75, 101] [9, 14, 20, 66, 72, 85, 90] [2, 5, 10, 64, 73, 114, 123] [3, 6, 11, 65, 67, 74, 111] [7, 12, 18, 65, 75, 110, 124] [8, 13, 19, 68, 76, 81, 89] [9, 14, 20, 69, 77, 78, 91] [15, 21, 27, 64, 70, 78, 109] [16, 22, 28, 71, 74, 79, 93] [17, 23, 29, 72, 80, 103, 122] [7, 12, 18, 68, 81, 98, 125] [8, 13, 19, 70, 73, 82, 95] [9, 14, 20, 74, 79, 83, 84] [15, 21, 27, 67, 75, 84, 97] [16, 22, 28, 66, 76, 85, 99] [17, 23, 29, 77, 86, 109, 114] [24, 30, 36, 77, 78, 82, 87] [25, 31, 37, 79, 88, 110, 111] [26, 32, 38, 80, 89, 99, 120] [15, 21, 27, 63, 81, 90, 103] [16, 22, 28, 69, 82, 87, 91] [17, 23, 29, 83, 92, 97, 119] [24, 30, 36, 71, 83, 84, 93] [25, 31, 37, 85, 94, 98, 116] [26, 32, 38, 86, 87, 95, 123] [33, 39, 45, 87, 95, 96, 117] [34, 40, 46, 83, 88, 97, 124] [35, 41, 47, 76, 89, 98, 125] [24, 30, 36, 72, 76, 90, 99] [25, 31, 37, 91, 100, 104, 105] [26, 32, 38, 92, 93, 101, 115] [33, 39, 45, 93, 101, 102, 106] [34, 40, 46, 72, 94, 103, 125] [35, 41, 47, 82, 95, 104, 121] [42, 48, 53, 91, 96, 105, 113] [43, 49, 54, 84, 97, 106, 118] [44, 50, 55, 94, 98, 107, 108] [33, 39, 45, 89, 99, 108, 112] [34, 40, 46, 77, 100, 109, 121] [35, 41, 47, 71, 101, 110, 124] [42, 48, 53, 79, 102, 111, 118] [43, 49, 54, 90, 103, 107, 112] [44, 50, 55, 96, 100, 104, 113] [51, 56, 59, 69, 105, 114, 123] [52, 57, 60, 92, 102, 106, 115] [42, 48, 53, 85, 107, 108, 116] [43, 49, 54, 78, 109, 113, 117] [44, 50, 55, 88, 102, 110, 118] [51, 56, 59, 74, 111, 115, 119] [52, 57, 60, 80, 108, 112, 120] [58, 61, 62, 86, 100, 114, 121] [51, 56, 59, 66, 116, 120, 122] [52, 57, 60, 86, 96, 117, 123] [58, 61, 62, 88, 92, 119, 124] [58, 61, 62, 80, 94, 122, 125]
H_Z (63 checks, sparse supports)
[0, 3, 5, 27, 63, 64, 67] [1, 6, 10, 15, 63, 64, 67] [2, 8, 11, 12, 65, 68, 73] [3, 9, 22, 59, 66, 69, 74] [2, 4, 11, 21, 63, 64, 67] [3, 5, 13, 18, 65, 68, 73] [6, 14, 28, 51, 66, 69, 74] [1, 7, 15, 19, 70, 75, 81] [8, 16, 30, 47, 71, 76, 82] [9, 17, 36, 40, 72, 77, 83] [6, 7, 10, 19, 65, 68, 73] [11, 16, 20, 56, 66, 69, 74] [4, 8, 12, 21, 70, 75, 81] [13, 22, 35, 36, 71, 76, 82] [14, 23, 24, 46, 72, 77, 83] [14, 15, 24, 54, 78, 84, 90] [16, 20, 25, 48, 79, 85, 91] [17, 26, 57, 62, 80, 86, 92] [0, 13, 18, 27, 70, 75, 81] [19, 24, 28, 41, 71, 76, 82] [20, 29, 30, 34, 72, 77, 83] [20, 21, 30, 43, 78, 84, 90] [9, 22, 31, 53, 79, 85, 91] [23, 32, 58, 60, 80, 86, 92] [24, 28, 32, 33, 87, 93, 99] [25, 34, 55, 61, 88, 94, 100] [13, 26, 35, 45, 89, 95, 101] [9, 27, 36, 49, 78, 84, 90] [14, 28, 37, 42, 79, 85, 91] [29, 38, 52, 61, 80, 86, 92] [16, 30, 38, 39, 87, 93, 99] [31, 40, 44, 62, 88, 94, 100] [19, 32, 33, 41, 89, 95, 101] [33, 42, 50, 60, 96, 102, 108] [21, 29, 34, 43, 97, 103, 109] [18, 31, 35, 44, 98, 104, 110] [22, 26, 36, 45, 87, 93, 99] [37, 46, 50, 58, 88, 94, 100] [8, 38, 39, 47, 89, 95, 101] [39, 48, 52, 55, 96, 102, 108] [17, 27, 40, 49, 97, 103, 109] [7, 37, 41, 50, 98, 104, 110] [6, 37, 42, 51, 105, 111, 116] [4, 39, 43, 52, 106, 112, 117] [0, 44, 49, 53, 107, 113, 118] [44, 45, 53, 57, 96, 102, 108] [15, 23, 46, 54, 97, 103, 109] [12, 25, 47, 55, 98, 104, 110] [11, 25, 48, 56, 105, 111, 116] [0, 45, 49, 57, 106, 112, 117] [1, 42, 50, 54, 107, 113, 118] [10, 23, 51, 58, 114, 119, 122] [2, 38, 52, 56, 115, 120, 123] [3, 31, 53, 59, 105, 111, 116] [1, 33, 54, 60, 106, 112, 117] [4, 43, 48, 55, 107, 113, 118] [2, 29, 56, 61, 114, 119, 122] [5, 26, 57, 59, 115, 120, 123] [7, 41, 46, 58, 121, 124, 125] [5, 17, 59, 62, 114, 119, 122] [10, 32, 51, 60, 115, 120, 123] [12, 34, 47, 61, 121, 124, 125] [18, 35, 40, 62, 121, 124, 125]
Code ID 126-40-2 · download JSON · raw on GitHub