← back to the board
[[126,4,15]] d ≤
n
126
k
4
d
15
kd²/n
7.143
w
7
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[1, 4, 12, 20, 39, 57, 59, 63, 64, 87, 93, 96, 102, 114, 119]
d_Z 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[1, 15, 20, 21, 27, 42, 47, 58, 65, 79, 80, 87, 98, 103, 120]
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 (3,4): 378 (3,5): 567 (3,6): 5943 (3,7): 9891 (3,8): 2205 (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 (3,4): 378 (3,5): 567 (3,6): 5943 (3,7): 9891 (3,8): 2205 (3,9): 1134 (3,10): 252

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_k4.txt, line 15. nonidentity supports a=[2, 20], b=[3, 16, 36].
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

[[126,4,15]] — 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 = 7.14.

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_k4.txt, line 15. This row is SmallGroup(63,3) with nonidentity GAP supports a=[2, 20], b=[3, 16, 36]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 15, not an exact certificate. The published distance (d=15) 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, 20], b=[3, 16, 36] 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) = 4. 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, 4, 56, 63, 70, 72, 115] [0, 1, 45, 64, 75, 77, 120] [2, 10, 41, 63, 65, 79, 121] [3, 11, 61, 65, 66, 78, 80] [1, 4, 58, 67, 81, 83, 123] [2, 5, 29, 64, 68, 85, 124] [3, 6, 53, 68, 69, 84, 86] [7, 18, 49, 65, 70, 87, 107] [8, 19, 50, 66, 70, 71, 88] [9, 12, 20, 71, 72, 87, 89] [5, 10, 44, 67, 73, 91, 125] [6, 7, 11, 73, 74, 90, 92] [7, 12, 37, 68, 75, 93, 113] [8, 13, 38, 69, 75, 76, 94] [9, 14, 59, 76, 77, 93, 95] [15, 27, 57, 63, 71, 78, 96] [1, 16, 28, 72, 78, 79, 97] [17, 21, 29, 79, 80, 96, 98] [12, 18, 52, 73, 81, 99, 118] [0, 13, 19, 74, 81, 82, 100] [14, 15, 20, 82, 83, 99, 101] [15, 21, 46, 64, 76, 84, 102] [16, 22, 47, 77, 84, 85, 103] [17, 23, 62, 85, 86, 102, 104] [5, 24, 36, 66, 79, 87, 105] [6, 25, 37, 80, 87, 88, 106] [26, 30, 38, 88, 89, 105, 107] [2, 21, 27, 67, 82, 90, 108] [3, 22, 28, 83, 90, 91, 109] [23, 24, 29, 91, 92, 108, 110] [24, 30, 54, 69, 85, 93, 111] [25, 31, 55, 86, 93, 94, 112] [18, 26, 32, 94, 95, 111, 113] [13, 33, 45, 72, 88, 96, 114] [14, 34, 46, 89, 96, 97, 115] [35, 39, 47, 63, 97, 98, 114] [8, 30, 36, 74, 91, 99, 116] [9, 31, 37, 92, 99, 100, 117] [32, 33, 38, 100, 101, 116, 118] [33, 39, 60, 77, 94, 102, 119] [4, 34, 40, 95, 102, 103, 120] [27, 35, 41, 64, 103, 104, 119] [22, 42, 53, 80, 97, 105, 121] [23, 43, 54, 65, 98, 105, 106] [44, 48, 55, 66, 106, 107, 121] [16, 39, 45, 83, 100, 108, 122] [17, 40, 46, 101, 108, 109, 123] [41, 42, 47, 67, 109, 110, 122] [10, 42, 48, 86, 103, 111, 124] [11, 43, 49, 68, 104, 111, 112] [36, 44, 50, 69, 112, 113, 124] [31, 51, 59, 70, 89, 106, 114] [32, 52, 60, 71, 107, 114, 115] [25, 48, 53, 92, 109, 116, 125] [26, 49, 54, 73, 110, 116, 117] [50, 51, 55, 74, 117, 118, 125] [19, 51, 56, 75, 95, 112, 119] [20, 52, 57, 76, 113, 119, 120] [40, 58, 62, 78, 98, 115, 121] [34, 56, 59, 81, 101, 117, 122] [35, 57, 60, 82, 118, 122, 123] [28, 58, 61, 84, 104, 120, 124] [43, 61, 62, 90, 110, 123, 125]
H_Z (63 checks, sparse supports)
[0, 2, 15, 35, 63, 64, 82] [1, 5, 21, 41, 64, 67, 79] [2, 3, 7, 43, 65, 68, 90] [3, 8, 24, 44, 66, 69, 91] [4, 10, 27, 47, 63, 67, 103] [5, 6, 12, 49, 68, 73, 87] [6, 13, 30, 50, 69, 74, 88] [0, 7, 8, 51, 70, 74, 75] [8, 9, 15, 52, 71, 76, 99] [0, 9, 16, 33, 72, 77, 100] [10, 11, 18, 54, 65, 73, 111] [11, 19, 36, 55, 66, 74, 112] [1, 12, 13, 56, 72, 75, 81] [13, 14, 21, 57, 76, 82, 96] [1, 14, 22, 39, 77, 83, 97] [3, 15, 16, 58, 78, 83, 84] [2, 16, 17, 24, 79, 85, 108] [3, 17, 25, 42, 80, 86, 109] [4, 18, 19, 59, 70, 81, 95] [19, 20, 27, 60, 71, 82, 119] [4, 20, 28, 45, 72, 83, 120] [6, 21, 22, 61, 80, 84, 90] [5, 22, 23, 30, 85, 91, 105] [6, 23, 31, 48, 86, 92, 106] [7, 9, 24, 25, 87, 92, 93] [8, 25, 26, 33, 88, 94, 116] [9, 26, 34, 51, 89, 95, 117] [11, 27, 28, 62, 78, 90, 104] [10, 28, 29, 36, 79, 91, 124] [11, 29, 37, 53, 68, 80, 92] [12, 14, 30, 31, 89, 93, 99] [13, 31, 32, 39, 94, 100, 114] [14, 32, 40, 56, 95, 101, 115] [15, 17, 33, 34, 96, 101, 102] [16, 34, 35, 42, 97, 103, 122] [17, 35, 43, 58, 98, 104, 123] [18, 20, 36, 37, 87, 99, 113] [19, 37, 38, 45, 75, 88, 100] [20, 38, 46, 59, 76, 89, 101] [21, 23, 39, 40, 98, 102, 108] [22, 40, 41, 48, 103, 109, 121] [23, 41, 49, 61, 65, 104, 110] [24, 26, 42, 43, 105, 110, 111] [25, 43, 44, 51, 106, 112, 125] [7, 26, 44, 52, 73, 107, 113] [27, 29, 45, 46, 64, 96, 108] [28, 46, 47, 53, 84, 97, 109] [29, 47, 54, 62, 85, 98, 110] [30, 32, 48, 49, 107, 111, 116] [31, 49, 50, 56, 70, 112, 117] [12, 32, 50, 57, 71, 113, 118] [33, 35, 51, 52, 114, 118, 119] [0, 34, 52, 58, 81, 115, 120] [36, 38, 53, 54, 69, 105, 116] [37, 54, 55, 59, 93, 106, 117] [18, 38, 55, 60, 94, 107, 118] [39, 41, 56, 57, 63, 119, 122] [1, 40, 57, 61, 78, 120, 123] [2, 42, 44, 58, 67, 121, 124] [45, 47, 59, 60, 77, 114, 122] [4, 46, 60, 62, 102, 115, 123] [5, 48, 50, 61, 66, 124, 125] [10, 53, 55, 62, 86, 121, 125]
Code ID 126-4-15 · download JSON · raw on GitHub