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[[126,46,3]] d =
n
126
k
46
d
3
kd²/n
3.286
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 3, d_Z = 3 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[80, 86, 92]
d_Z 3 · witness weight 3 (claimed upper_bound)
witness operator (support, 3 qubits)
[38, 46, 53]
certificate exact, d = 3 · scipy/HiGHS MILP
X: no logical < 3 exists; Z: no logical < 3 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 8 · H_Z 8
qubit degrees H_X 4 · H_Z 4
trapping sets H_X (1,4)×126 (2,4)×504 (3,0)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 126 (2,4): 504 (2,6): 756 (3,0): 63 (3,4): 1953 (3,6): 6342 (3,8): 6048
trapping sets H_Z (1,4)×126 (2,4)×504 (3,0)×63 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 126 (2,4): 504 (2,6): 756 (3,0): 63 (3,4): 1953 (3,6): 6342 (3,8): 6048

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_wt8wtL4_order63_k46.txt, line 1. nonidentity supports a=[2, 3, 6], b=[3, 4, 9].
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

[[126,46,3]] — two-block group-algebra code on SmallGroup(63,3)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 3.29.

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_wt8wtL4_order63_k46.txt, line 1. This row is SmallGroup(63,3) with nonidentity GAP supports a=[2, 3, 6], b=[3, 4, 9]. 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=46), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 3, not an exact certificate. The published distance 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, 3, 6], b=[3, 4, 9] 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) = 46. 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 8) · Z-checks 63 (max weight 8)
H_X (63 checks, sparse supports)
[0, 4, 7, 18, 63, 70, 107, 121] [0, 1, 7, 12, 64, 75, 113, 124] [0, 2, 4, 10, 63, 65, 107, 115] [3, 11, 15, 27, 63, 66, 70, 78] [1, 4, 12, 18, 67, 81, 118, 125] [0, 1, 2, 5, 64, 68, 113, 120] [3, 6, 15, 21, 64, 69, 75, 84] [2, 7, 10, 18, 65, 70, 115, 121] [3, 8, 11, 19, 63, 65, 66, 71] [9, 20, 24, 36, 66, 72, 78, 87] [1, 4, 5, 10, 67, 73, 118, 123] [6, 11, 21, 27, 67, 74, 81, 90] [2, 5, 7, 12, 68, 75, 120, 124] [3, 6, 8, 13, 64, 68, 69, 76] [9, 14, 24, 30, 69, 77, 84, 93] [8, 15, 19, 27, 65, 70, 71, 78] [9, 16, 20, 28, 66, 71, 72, 79] [17, 29, 33, 45, 72, 80, 87, 96] [5, 10, 12, 18, 73, 81, 123, 125] [6, 11, 13, 19, 67, 73, 74, 82] [14, 20, 30, 36, 74, 83, 90, 99] [8, 13, 15, 21, 68, 75, 76, 84] [9, 14, 16, 22, 69, 76, 77, 85] [17, 23, 33, 39, 77, 86, 93, 102] [16, 24, 28, 36, 71, 78, 79, 87] [17, 25, 29, 37, 72, 79, 80, 88] [26, 38, 42, 53, 80, 89, 96, 105] [13, 19, 21, 27, 73, 81, 82, 90] [14, 20, 22, 28, 74, 82, 83, 91] [23, 29, 39, 45, 83, 92, 99, 108] [16, 22, 24, 30, 76, 84, 85, 93] [17, 23, 25, 31, 77, 85, 86, 94] [26, 32, 42, 48, 86, 95, 102, 111] [25, 33, 37, 45, 79, 87, 88, 96] [26, 34, 38, 46, 80, 88, 89, 97] [35, 47, 51, 59, 89, 98, 105, 114] [22, 28, 30, 36, 82, 90, 91, 99] [23, 29, 31, 37, 83, 91, 92, 100] [32, 38, 48, 53, 92, 101, 108, 116] [25, 31, 33, 39, 85, 93, 94, 102] [26, 32, 34, 40, 86, 94, 95, 103] [35, 41, 51, 56, 95, 104, 111, 119] [34, 42, 46, 53, 88, 96, 97, 105] [35, 43, 47, 54, 89, 97, 98, 106] [44, 55, 58, 62, 98, 107, 114, 121] [31, 37, 39, 45, 91, 99, 100, 108] [32, 38, 40, 46, 92, 100, 101, 109] [41, 47, 56, 59, 101, 110, 116, 122] [34, 40, 42, 48, 94, 102, 103, 111] [35, 41, 43, 49, 95, 103, 104, 112] [44, 50, 58, 61, 104, 113, 119, 124] [43, 51, 54, 59, 97, 105, 106, 114] [44, 52, 55, 60, 98, 106, 107, 115] [40, 46, 48, 53, 100, 108, 109, 116] [41, 47, 49, 54, 101, 109, 110, 117] [50, 55, 61, 62, 110, 118, 122, 125] [43, 49, 51, 56, 103, 111, 112, 119] [44, 50, 52, 57, 104, 112, 113, 120] [52, 58, 60, 62, 106, 114, 115, 121] [49, 54, 56, 59, 109, 116, 117, 122] [50, 55, 57, 60, 110, 117, 118, 123] [52, 57, 58, 61, 112, 119, 120, 124] [57, 60, 61, 62, 117, 122, 123, 125]
H_Z (63 checks, sparse supports)
[0, 2, 3, 8, 63, 64, 65, 68] [1, 5, 6, 13, 64, 67, 68, 73] [2, 7, 8, 15, 65, 68, 70, 75] [3, 8, 9, 16, 66, 69, 71, 76] [4, 10, 11, 19, 63, 65, 67, 73] [5, 12, 13, 21, 68, 73, 75, 81] [6, 13, 14, 22, 69, 74, 76, 82] [0, 3, 7, 15, 63, 64, 70, 75] [8, 15, 16, 24, 71, 76, 78, 84] [9, 16, 17, 25, 72, 77, 79, 85] [10, 18, 19, 27, 65, 70, 73, 81] [11, 19, 20, 28, 66, 71, 74, 82] [1, 6, 12, 21, 64, 67, 75, 81] [13, 21, 22, 30, 76, 82, 84, 90] [14, 22, 23, 31, 77, 83, 85, 91] [3, 9, 15, 24, 66, 69, 78, 84] [16, 24, 25, 33, 79, 85, 87, 93] [17, 25, 26, 34, 80, 86, 88, 94] [4, 11, 18, 27, 63, 67, 70, 81] [19, 27, 28, 36, 71, 78, 82, 90] [20, 28, 29, 37, 72, 79, 83, 91] [6, 14, 21, 30, 69, 74, 84, 90] [22, 30, 31, 39, 85, 91, 93, 99] [23, 31, 32, 40, 86, 92, 94, 100] [9, 17, 24, 33, 72, 77, 87, 93] [25, 33, 34, 42, 88, 94, 96, 102] [26, 34, 35, 43, 89, 95, 97, 103] [11, 20, 27, 36, 66, 74, 78, 90] [28, 36, 37, 45, 79, 87, 91, 99] [29, 37, 38, 46, 80, 88, 92, 100] [14, 23, 30, 39, 77, 83, 93, 99] [31, 39, 40, 48, 94, 100, 102, 108] [32, 40, 41, 49, 95, 101, 103, 109] [17, 26, 33, 42, 80, 86, 96, 102] [34, 42, 43, 51, 97, 103, 105, 111] [35, 43, 44, 52, 98, 104, 106, 112] [20, 29, 36, 45, 72, 83, 87, 99] [37, 45, 46, 53, 88, 96, 100, 108] [38, 46, 47, 54, 89, 97, 101, 109] [23, 32, 39, 48, 86, 92, 102, 108] [40, 48, 49, 56, 103, 109, 111, 116] [41, 49, 50, 57, 104, 110, 112, 117] [26, 35, 42, 51, 89, 95, 105, 111] [43, 51, 52, 58, 106, 112, 114, 119] [0, 2, 44, 52, 107, 113, 115, 120] [29, 38, 45, 53, 80, 92, 96, 108] [46, 53, 54, 59, 97, 105, 109, 116] [47, 54, 55, 60, 98, 106, 110, 117] [32, 41, 48, 56, 95, 101, 111, 116] [49, 56, 57, 61, 112, 117, 119, 122] [1, 5, 50, 57, 113, 118, 120, 123] [35, 44, 51, 58, 98, 104, 114, 119] [2, 7, 52, 58, 115, 120, 121, 124] [38, 47, 53, 59, 89, 101, 105, 116] [54, 59, 60, 62, 106, 114, 117, 122] [4, 10, 55, 60, 107, 115, 118, 123] [41, 50, 56, 61, 104, 110, 119, 122] [5, 12, 57, 61, 120, 123, 124, 125] [0, 7, 44, 58, 107, 113, 121, 124] [47, 55, 59, 62, 98, 110, 114, 122] [10, 18, 60, 62, 115, 121, 123, 125] [1, 12, 50, 61, 113, 118, 124, 125] [4, 18, 55, 62, 107, 118, 121, 125]
Code ID 126-46-3 · download JSON · raw on GitHub