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[[126,36,2]] d =
n
126
k
36
d
2
kd²/n
1.143
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 (2,3): 27 (2,4): 63 (2,5): 702 (2,6): 252 (3,0): 21 (3,3): 48 (3,4): 891 (3,5): 1161 (3,6): 3690 (3,7): 6453 (3,8): 1197 (3,9): 675
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 (2,3): 27 (2,4): 63 (2,5): 702 (2,6): 252 (3,0): 21 (3,3): 48 (3,4): 891 (3,5): 1161 (3,6): 3690 (3,7): 6453 (3,8): 1197 (3,9): 675

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_k36.txt, line 1. nonidentity supports a=[2, 5], b=[2, 20, 46].
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

[[126,36,2]] — 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 = 1.14.

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_k36.txt, line 1. This row is SmallGroup(63,3) with nonidentity GAP supports a=[2, 5], b=[2, 20, 46]. 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=36), 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=[2, 20, 46] 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) = 36. 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, 67, 76, 119] [0, 1, 4, 63, 64, 82, 122] [2, 5, 10, 65, 73, 84, 104] [3, 6, 11, 66, 75, 94, 101] [0, 1, 4, 64, 67, 71, 114] [2, 5, 10, 65, 68, 90, 110] [3, 6, 11, 69, 81, 89, 100] [7, 12, 18, 69, 70, 81, 112] [8, 13, 19, 64, 71, 102, 109] [9, 14, 20, 72, 74, 93, 112] [2, 5, 10, 68, 73, 78, 98] [3, 6, 11, 70, 74, 88, 95] [7, 12, 18, 70, 74, 75, 117] [8, 13, 19, 67, 76, 97, 108] [9, 14, 20, 66, 77, 99, 117] [15, 21, 27, 68, 78, 86, 116] [16, 22, 28, 77, 79, 82, 119] [17, 23, 29, 68, 80, 110, 111] [7, 12, 18, 66, 75, 81, 106] [8, 13, 19, 63, 82, 96, 103] [9, 14, 20, 69, 83, 87, 106] [15, 21, 27, 73, 84, 92, 105] [16, 22, 28, 71, 83, 85, 122] [17, 23, 29, 73, 86, 98, 116] [24, 30, 36, 85, 87, 90, 104] [25, 31, 37, 75, 88, 95, 117] [26, 32, 38, 83, 85, 89, 124] [15, 21, 27, 65, 80, 90, 111] [16, 22, 28, 72, 76, 91, 114] [17, 23, 29, 65, 92, 104, 105] [24, 30, 36, 78, 91, 93, 110] [25, 31, 37, 81, 94, 101, 106] [26, 32, 38, 72, 91, 95, 125] [33, 39, 45, 64, 96, 103, 122] [34, 40, 46, 91, 93, 97, 113] [35, 41, 47, 84, 98, 103, 118] [24, 30, 36, 79, 84, 98, 99] [25, 31, 37, 70, 89, 100, 112] [26, 32, 38, 77, 79, 101, 121] [33, 39, 45, 67, 102, 109, 114] [34, 40, 46, 79, 99, 103, 118] [35, 41, 47, 90, 104, 107, 109] [42, 48, 53, 77, 99, 105, 120] [43, 49, 54, 69, 106, 111, 123] [44, 50, 55, 92, 102, 107, 120] [33, 39, 45, 63, 97, 108, 119] [34, 40, 46, 85, 87, 107, 109] [35, 41, 47, 78, 97, 110, 113] [42, 48, 53, 83, 87, 111, 123] [43, 49, 54, 74, 112, 115, 116] [44, 50, 55, 80, 108, 113, 123] [51, 56, 59, 76, 95, 114, 125] [52, 57, 60, 86, 100, 115, 124] [42, 48, 53, 72, 93, 115, 116] [43, 49, 54, 66, 105, 117, 120] [44, 50, 55, 86, 96, 115, 118] [51, 56, 59, 82, 101, 119, 121] [52, 57, 60, 88, 92, 120, 125] [58, 61, 62, 94, 108, 113, 121] [51, 56, 59, 71, 89, 122, 124] [52, 57, 60, 80, 94, 121, 123] [58, 61, 62, 96, 100, 118, 124] [58, 61, 62, 88, 102, 107, 125]
H_Z (63 checks, sparse supports)
[0, 1, 19, 45, 63, 64, 67] [1, 4, 8, 33, 63, 64, 67] [2, 5, 27, 29, 65, 68, 73] [3, 14, 18, 54, 66, 69, 74] [0, 4, 13, 39, 63, 64, 67] [5, 10, 15, 17, 65, 68, 73] [6, 7, 20, 43, 66, 69, 74] [7, 11, 12, 37, 70, 75, 81] [4, 8, 22, 59, 71, 76, 82] [9, 28, 32, 53, 72, 77, 83] [2, 10, 21, 23, 65, 68, 73] [9, 11, 12, 49, 66, 69, 74] [3, 12, 18, 25, 70, 75, 81] [0, 13, 28, 51, 71, 76, 82] [14, 16, 38, 42, 72, 77, 83] [10, 15, 30, 47, 78, 84, 90] [16, 36, 38, 40, 79, 85, 91] [17, 27, 50, 60, 80, 86, 92] [6, 7, 18, 31, 70, 75, 81] [1, 16, 19, 56, 71, 76, 82] [20, 22, 26, 48, 72, 77, 83] [2, 21, 35, 36, 78, 84, 90] [22, 24, 26, 46, 79, 85, 91] [15, 23, 52, 55, 80, 86, 92] [20, 24, 46, 48, 87, 93, 99] [11, 25, 57, 62, 88, 94, 100] [6, 26, 37, 59, 89, 95, 101] [5, 24, 27, 41, 78, 84, 90] [28, 30, 32, 34, 79, 85, 91] [21, 29, 44, 57, 80, 86, 92] [9, 30, 34, 53, 87, 93, 99] [3, 31, 58, 60, 88, 94, 100] [11, 25, 32, 51, 89, 95, 101] [19, 33, 55, 61, 96, 102, 108] [13, 34, 45, 47, 97, 103, 109] [10, 23, 35, 36, 98, 104, 110] [14, 36, 40, 42, 87, 93, 99] [6, 37, 52, 61, 88, 94, 100] [3, 31, 38, 56, 89, 95, 101] [8, 39, 44, 62, 96, 102, 108] [19, 33, 35, 40, 97, 103, 109] [2, 24, 29, 41, 98, 104, 110] [21, 29, 42, 54, 105, 111, 116] [18, 20, 31, 43, 106, 112, 117] [41, 44, 46, 62, 107, 113, 118] [13, 45, 50, 58, 96, 102, 108] [8, 39, 41, 46, 97, 103, 109] [5, 17, 30, 47, 98, 104, 110] [17, 27, 43, 48, 105, 111, 116] [7, 9, 37, 49, 106, 112, 117] [34, 47, 50, 58, 107, 113, 118] [4, 28, 39, 51, 114, 119, 122] [49, 52, 53, 55, 115, 120, 123] [15, 23, 49, 53, 105, 111, 116] [12, 14, 25, 54, 106, 112, 117] [35, 40, 55, 61, 107, 113, 118] [0, 16, 45, 56, 114, 119, 122] [42, 44, 54, 57, 115, 120, 123] [38, 56, 58, 60, 121, 124, 125] [1, 22, 33, 59, 114, 119, 122] [43, 48, 50, 60, 115, 120, 123] [26, 52, 59, 61, 121, 124, 125] [32, 51, 57, 62, 121, 124, 125]
Code ID 126-36-2 · download JSON · raw on GitHub