← back to the board
[[132,4,15]] d ≤
n
132
k
4
d
15
kd²/n
6.818
w
7
X/Z
1.07

Share this result

Distance

X/Z asymmetry 1.07 · d_X ≤ 16, 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 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[11, 12, 22, 27, 37, 46, 48, 57, 70, 81, 85, 89, 95, 102, 125, 131]
d_Z 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[8, 12, 13, 19, 32, 61, 67, 76, 79, 106, 116, 118, 121, 123, 125]
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)×66 (2,4)×198 (3,3)×88 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 66 (1,4): 66 (2,4): 198 (2,5): 792 (2,6): 396 (3,3): 88 (3,4): 66 (3,5): 924 (3,6): 6600 (3,7): 9174 (3,8): 4158 (3,9): 1188 (3,10): 264
trapping sets H_Z (1,3)×66 (2,4)×198 (3,3)×88 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 66 (1,4): 66 (2,4): 198 (2,5): 792 (2,6): 396 (3,3): 88 (3,4): 66 (3,5): 924 (3,6): 6600 (3,7): 9174 (3,8): 4158 (3,9): 1188 (3,10): 264

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. Published d=16, but the gate found d=15; honest claim is d<=15.
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

[[132,4,15]] — two-block group-algebra code on SmallGroup(66,1)

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 = 4·225/132 = 6.82.

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_order66_k4.txt, line 9. This row is SmallGroup(66,1) with nonidentity GAP supports a=[4,15], b=[5,14,36]. 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. Note: the published distance is d=16, but the gate found a lighter logical of weight 15, so the honest claim here is d ≤ 15 (the published value was not supported by the witness search).

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(66,1), 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=[4,15] and b=[5,14,36] 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 = 132 - 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 — the gate found d=15, not the published d=16.

Parity checks

X-checks 66 (max weight 7) · Z-checks 66 (max weight 7)
H_X (66 checks, sparse supports)
[0, 8, 60, 66, 119, 123, 127] [1, 5, 65, 67, 116, 120, 129] [2, 3, 14, 67, 68, 125, 130] [0, 3, 64, 69, 112, 114, 131] [4, 10, 11, 66, 70, 122, 131] [5, 11, 57, 71, 109, 117, 130] [6, 7, 20, 70, 72, 74, 129] [2, 7, 8, 73, 77, 118, 120] [3, 8, 54, 74, 105, 121, 128] [9, 16, 17, 68, 75, 77, 127] [1, 10, 17, 74, 76, 115, 123] [1, 11, 62, 77, 102, 124, 126] [12, 13, 26, 75, 78, 80, 131] [6, 13, 14, 66, 79, 83, 124] [0, 7, 14, 71, 80, 111, 126] [15, 22, 23, 72, 81, 83, 130] [4, 16, 23, 67, 80, 82, 121] [4, 5, 17, 69, 83, 108, 128] [18, 19, 32, 77, 81, 84, 86] [12, 19, 20, 68, 85, 89, 128] [2, 13, 20, 69, 76, 86, 117] [21, 28, 29, 74, 78, 87, 89] [9, 22, 29, 70, 86, 88, 126] [9, 10, 23, 71, 73, 89, 114] [24, 25, 38, 83, 87, 90, 92] [18, 25, 26, 71, 72, 91, 95] [6, 19, 26, 73, 82, 92, 123] [27, 34, 35, 80, 84, 93, 95] [15, 28, 35, 69, 75, 92, 94] [15, 16, 29, 76, 79, 95, 120] [30, 31, 44, 89, 93, 96, 98] [24, 31, 32, 76, 78, 97, 101] [12, 25, 32, 67, 79, 88, 98] [33, 40, 41, 86, 90, 99, 101] [21, 34, 41, 73, 81, 98, 100] [21, 22, 35, 66, 82, 85, 101] [36, 37, 50, 95, 99, 102, 104] [30, 37, 38, 82, 84, 103, 107] [18, 31, 38, 70, 85, 94, 104] [39, 46, 47, 92, 96, 105, 107] [27, 40, 47, 79, 87, 104, 106] [27, 28, 41, 68, 88, 91, 107] [42, 43, 56, 101, 105, 108, 110] [36, 43, 44, 88, 90, 109, 113] [24, 37, 44, 75, 91, 100, 110] [45, 52, 53, 98, 102, 111, 113] [33, 46, 53, 85, 93, 110, 112] [33, 34, 47, 72, 94, 97, 113] [48, 49, 61, 107, 111, 114, 116] [42, 49, 50, 94, 96, 115, 119] [30, 43, 50, 81, 97, 106, 116] [51, 58, 59, 104, 108, 117, 119] [39, 52, 59, 91, 99, 116, 118] [39, 40, 53, 78, 100, 103, 119] [54, 55, 64, 113, 117, 120, 122] [48, 55, 56, 100, 102, 121, 125] [36, 49, 56, 87, 103, 112, 122] [57, 62, 63, 110, 114, 123, 125] [45, 58, 63, 97, 105, 122, 124] [45, 46, 59, 84, 106, 109, 125] [54, 60, 61, 106, 108, 126, 129] [42, 55, 61, 93, 109, 118, 127] [51, 62, 65, 103, 111, 127, 128] [51, 52, 63, 90, 112, 115, 129] [48, 60, 64, 99, 115, 124, 130] [57, 58, 65, 96, 118, 121, 131]
H_Z (66 checks, sparse supports)
[0, 4, 13, 35, 66, 69, 80] [1, 2, 16, 32, 67, 76, 77] [2, 9, 19, 41, 68, 73, 86] [3, 17, 20, 28, 68, 69, 74] [4, 6, 22, 38, 70, 82, 83] [5, 14, 23, 25, 67, 71, 83] [6, 15, 25, 47, 72, 79, 92] [7, 23, 26, 34, 72, 73, 80] [6, 8, 10, 21, 66, 73, 74] [9, 12, 28, 44, 75, 88, 89] [10, 20, 29, 31, 70, 76, 89] [7, 9, 11, 18, 70, 71, 77] [12, 21, 31, 53, 78, 85, 98] [13, 29, 32, 40, 78, 79, 86] [12, 14, 16, 27, 68, 79, 80] [15, 18, 34, 50, 81, 94, 95] [16, 26, 35, 37, 75, 82, 95] [13, 15, 17, 24, 75, 76, 83] [18, 27, 37, 59, 84, 91, 104] [19, 35, 38, 46, 84, 85, 92] [18, 20, 22, 33, 72, 85, 86] [21, 24, 40, 56, 87, 100, 101] [22, 32, 41, 43, 81, 88, 101] [19, 21, 23, 30, 81, 82, 89] [24, 33, 43, 63, 90, 97, 110] [25, 41, 44, 52, 90, 91, 98] [24, 26, 28, 39, 78, 91, 92] [27, 30, 46, 61, 93, 106, 107] [28, 38, 47, 49, 87, 94, 107] [25, 27, 29, 36, 87, 88, 95] [30, 39, 49, 65, 96, 103, 116] [31, 47, 50, 58, 96, 97, 104] [30, 32, 34, 45, 84, 97, 98] [33, 36, 52, 64, 99, 112, 113] [34, 44, 53, 55, 93, 100, 113] [31, 33, 35, 42, 93, 94, 101] [11, 36, 45, 55, 102, 109, 122] [37, 53, 56, 62, 102, 103, 110] [36, 38, 40, 51, 90, 103, 104] [8, 39, 42, 58, 105, 118, 119] [40, 50, 59, 60, 99, 106, 119] [37, 39, 41, 48, 99, 100, 107] [17, 42, 51, 60, 108, 115, 127] [5, 43, 59, 61, 108, 109, 116] [42, 44, 46, 57, 96, 109, 110] [14, 45, 48, 62, 111, 124, 125] [3, 46, 56, 63, 105, 112, 125] [43, 45, 47, 54, 105, 106, 113] [3, 23, 48, 57, 114, 121, 130] [10, 49, 63, 64, 114, 115, 122] [1, 48, 50, 52, 102, 115, 116] [5, 20, 51, 54, 117, 128, 129] [7, 52, 61, 65, 111, 118, 129] [0, 49, 51, 53, 111, 112, 119] [1, 7, 29, 54, 74, 120, 126] [8, 16, 55, 65, 120, 121, 127] [4, 54, 56, 58, 108, 121, 122] [0, 10, 26, 57, 71, 123, 131] [11, 13, 58, 64, 117, 124, 131] [2, 55, 57, 59, 117, 118, 125] [11, 14, 22, 60, 66, 126, 130] [0, 9, 61, 62, 114, 126, 127] [8, 17, 19, 62, 77, 123, 128] [1, 6, 60, 63, 123, 124, 129] [2, 5, 15, 64, 69, 120, 130] [3, 4, 12, 65, 67, 128, 131]
Code ID 132-4-15 · download JSON · raw on GitHub