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[[132,44,2]] d =
n
132
k
44
d
2
kd²/n
1.333
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)
[84, 91]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[54, 64]
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)×66 (2,0)×99 (3,3)×418 (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 (2,0): 99 (2,5): 792 (2,6): 264 (3,3): 418 (3,4): 1056 (3,5): 792 (3,6): 3564 (3,7): 5544 (3,8): 1584 (3,9): 792
trapping sets H_Z (1,3)×66 (2,0)×66 (3,3)×418 (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 (2,0): 66 (2,4): 198 (2,5): 792 (3,3): 418 (3,4): 1386 (3,5): 1584 (3,6): 3564 (3,7): 3960 (3,8): 132

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(66,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt7wtL3_order66_k44.txt, line 1. nonidentity supports a=[4, 9], b=[2, 8, 11].
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

[[132,44,2]] — 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 = 1.33.

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_k44.txt, line 1. This row is SmallGroup(66,1) with nonidentity GAP supports a=[4, 9], b=[2, 8, 11]. 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=44), 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(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, 9], b=[2, 8, 11] 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) = 44. 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 66 (max weight 7) · Z-checks 66 (max weight 7)
H_X (66 checks, sparse supports)
[0, 3, 8, 66, 67, 128, 130] [1, 5, 11, 66, 67, 126, 131] [2, 7, 14, 68, 70, 71, 74] [0, 3, 8, 69, 77, 120, 123] [4, 10, 17, 68, 69, 70, 77] [1, 5, 11, 71, 74, 120, 123] [6, 13, 20, 72, 75, 76, 80] [2, 7, 14, 66, 67, 73, 83] [0, 3, 8, 71, 74, 126, 131] [9, 16, 23, 72, 73, 75, 83] [4, 10, 17, 66, 67, 76, 80] [1, 5, 11, 69, 77, 128, 130] [12, 19, 26, 78, 81, 82, 86] [6, 13, 20, 68, 70, 79, 89] [2, 7, 14, 69, 76, 77, 80] [15, 22, 29, 78, 79, 81, 89] [9, 16, 23, 68, 70, 82, 86] [4, 10, 17, 71, 73, 74, 83] [18, 25, 32, 84, 87, 88, 92] [12, 19, 26, 72, 75, 85, 95] [6, 13, 20, 73, 82, 83, 86] [21, 28, 35, 84, 85, 87, 95] [15, 22, 29, 72, 75, 88, 92] [9, 16, 23, 76, 79, 80, 89] [24, 31, 38, 90, 93, 94, 98] [18, 25, 32, 78, 81, 91, 101] [12, 19, 26, 79, 88, 89, 92] [27, 34, 41, 90, 91, 93, 101] [21, 28, 35, 78, 81, 94, 98] [15, 22, 29, 82, 85, 86, 95] [30, 37, 44, 96, 99, 100, 104] [24, 31, 38, 84, 87, 97, 107] [18, 25, 32, 85, 94, 95, 98] [33, 40, 47, 96, 97, 99, 107] [27, 34, 41, 84, 87, 100, 104] [21, 28, 35, 88, 91, 92, 101] [36, 43, 50, 102, 105, 106, 110] [30, 37, 44, 90, 93, 103, 113] [24, 31, 38, 91, 100, 101, 104] [39, 46, 53, 102, 103, 105, 113] [33, 40, 47, 90, 93, 106, 110] [27, 34, 41, 94, 97, 98, 107] [42, 49, 56, 108, 111, 112, 116] [36, 43, 50, 96, 99, 109, 119] [30, 37, 44, 97, 106, 107, 110] [45, 52, 59, 108, 109, 111, 119] [39, 46, 53, 96, 99, 112, 116] [33, 40, 47, 100, 103, 104, 113] [48, 55, 61, 114, 117, 118, 122] [42, 49, 56, 102, 105, 115, 125] [36, 43, 50, 103, 112, 113, 116] [51, 58, 63, 114, 115, 117, 125] [45, 52, 59, 102, 105, 118, 122] [39, 46, 53, 106, 109, 110, 119] [54, 60, 64, 120, 123, 124, 127] [48, 55, 61, 108, 111, 121, 129] [42, 49, 56, 109, 118, 119, 122] [57, 62, 65, 120, 121, 123, 129] [51, 58, 63, 108, 111, 124, 127] [45, 52, 59, 112, 115, 116, 125] [54, 60, 64, 114, 117, 126, 131] [48, 55, 61, 115, 124, 125, 127] [57, 62, 65, 114, 117, 128, 130] [51, 58, 63, 118, 121, 122, 129] [54, 60, 64, 121, 128, 129, 130] [57, 62, 65, 124, 126, 127, 131]
H_Z (66 checks, sparse supports)
[0, 1, 7, 10, 66, 69, 74] [0, 1, 7, 10, 67, 71, 77] [2, 4, 13, 16, 68, 73, 80] [3, 4, 11, 14, 66, 69, 74] [2, 4, 13, 16, 70, 76, 83] [2, 5, 8, 17, 67, 71, 77] [6, 9, 19, 22, 72, 79, 86] [7, 9, 17, 20, 68, 73, 80] [2, 5, 8, 17, 66, 69, 74] [6, 9, 19, 22, 75, 82, 89] [6, 10, 14, 23, 70, 76, 83] [3, 4, 11, 14, 67, 71, 77] [12, 15, 25, 28, 78, 85, 92] [13, 15, 23, 26, 72, 79, 86] [6, 10, 14, 23, 68, 73, 80] [12, 15, 25, 28, 81, 88, 95] [12, 16, 20, 29, 75, 82, 89] [7, 9, 17, 20, 70, 76, 83] [18, 21, 31, 34, 84, 91, 98] [19, 21, 29, 32, 78, 85, 92] [12, 16, 20, 29, 72, 79, 86] [18, 21, 31, 34, 87, 94, 101] [18, 22, 26, 35, 81, 88, 95] [13, 15, 23, 26, 75, 82, 89] [24, 27, 37, 40, 90, 97, 104] [25, 27, 35, 38, 84, 91, 98] [18, 22, 26, 35, 78, 85, 92] [24, 27, 37, 40, 93, 100, 107] [24, 28, 32, 41, 87, 94, 101] [19, 21, 29, 32, 81, 88, 95] [30, 33, 43, 46, 96, 103, 110] [31, 33, 41, 44, 90, 97, 104] [24, 28, 32, 41, 84, 91, 98] [30, 33, 43, 46, 99, 106, 113] [30, 34, 38, 47, 93, 100, 107] [25, 27, 35, 38, 87, 94, 101] [36, 39, 49, 52, 102, 109, 116] [37, 39, 47, 50, 96, 103, 110] [30, 34, 38, 47, 90, 97, 104] [36, 39, 49, 52, 105, 112, 119] [36, 40, 44, 53, 99, 106, 113] [31, 33, 41, 44, 93, 100, 107] [42, 45, 55, 58, 108, 115, 122] [43, 45, 53, 56, 102, 109, 116] [36, 40, 44, 53, 96, 103, 110] [42, 45, 55, 58, 111, 118, 125] [42, 46, 50, 59, 105, 112, 119] [37, 39, 47, 50, 99, 106, 113] [48, 51, 60, 62, 114, 121, 127] [49, 51, 59, 61, 108, 115, 122] [42, 46, 50, 59, 102, 109, 116] [48, 51, 60, 62, 117, 124, 129] [48, 52, 56, 63, 111, 118, 125] [43, 45, 53, 56, 105, 112, 119] [3, 5, 54, 57, 120, 126, 130] [55, 57, 63, 64, 114, 121, 127] [48, 52, 56, 63, 108, 115, 122] [3, 5, 54, 57, 123, 128, 131] [54, 58, 61, 65, 117, 124, 129] [49, 51, 59, 61, 111, 118, 125] [1, 8, 60, 65, 120, 126, 130] [54, 58, 61, 65, 114, 121, 127] [0, 11, 62, 64, 123, 128, 131] [55, 57, 63, 64, 117, 124, 129] [0, 11, 62, 64, 120, 126, 130] [1, 8, 60, 65, 123, 128, 131]
Code ID 132-44-2 · download JSON · raw on GitHub