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[[132,48,2]] d =
n
132
k
48
d
2
kd²/n
1.455
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)
[27, 41]
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 (2,4): 132 (2,5): 792 (3,3): 418 (3,4): 1188 (3,5): 1584 (3,6): 3564 (3,7): 3960
trapping sets H_Z (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_Z
(1,3): 66 (1,4): 66 (2,0): 99 (2,4): 132 (2,5): 792 (3,3): 418 (3,4): 1188 (3,5): 1584 (3,6): 3564 (3,7): 3960

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_k48.txt, line 1. nonidentity supports a=[4, 9], b=[2, 7, 10].
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

[[132,48,2]] — 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 = 1.45.

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_k48.txt, line 1. This row is SmallGroup(66,1) with nonidentity GAP supports a=[4, 9], b=[2, 7, 10]. 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=48), 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, 7, 10] 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) = 48. 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, 114, 117] [1, 5, 11, 66, 67, 114, 117] [2, 7, 14, 68, 70, 120, 123] [0, 3, 8, 69, 77, 121, 129] [4, 10, 17, 68, 70, 120, 123] [1, 5, 11, 71, 74, 124, 127] [6, 13, 20, 66, 67, 72, 75] [2, 7, 14, 73, 83, 126, 131] [0, 3, 8, 71, 74, 124, 127] [9, 16, 23, 66, 67, 72, 75] [4, 10, 17, 76, 80, 128, 130] [1, 5, 11, 69, 77, 121, 129] [12, 19, 26, 68, 70, 78, 81] [6, 13, 20, 69, 77, 79, 89] [2, 7, 14, 76, 80, 128, 130] [15, 22, 29, 68, 70, 78, 81] [9, 16, 23, 71, 74, 82, 86] [4, 10, 17, 73, 83, 126, 131] [18, 25, 32, 72, 75, 84, 87] [12, 19, 26, 73, 83, 85, 95] [6, 13, 20, 71, 74, 82, 86] [21, 28, 35, 72, 75, 84, 87] [15, 22, 29, 76, 80, 88, 92] [9, 16, 23, 69, 77, 79, 89] [24, 31, 38, 78, 81, 90, 93] [18, 25, 32, 79, 89, 91, 101] [12, 19, 26, 76, 80, 88, 92] [27, 34, 41, 78, 81, 90, 93] [21, 28, 35, 82, 86, 94, 98] [15, 22, 29, 73, 83, 85, 95] [30, 37, 44, 84, 87, 96, 99] [24, 31, 38, 85, 95, 97, 107] [18, 25, 32, 82, 86, 94, 98] [33, 40, 47, 84, 87, 96, 99] [27, 34, 41, 88, 92, 100, 104] [21, 28, 35, 79, 89, 91, 101] [36, 43, 50, 90, 93, 102, 105] [30, 37, 44, 91, 101, 103, 113] [24, 31, 38, 88, 92, 100, 104] [39, 46, 53, 90, 93, 102, 105] [33, 40, 47, 94, 98, 106, 110] [27, 34, 41, 85, 95, 97, 107] [42, 49, 56, 96, 99, 108, 111] [36, 43, 50, 97, 107, 109, 119] [30, 37, 44, 94, 98, 106, 110] [45, 52, 59, 96, 99, 108, 111] [39, 46, 53, 100, 104, 112, 116] [33, 40, 47, 91, 101, 103, 113] [48, 55, 61, 102, 105, 114, 117] [42, 49, 56, 103, 113, 115, 125] [36, 43, 50, 100, 104, 112, 116] [51, 58, 63, 102, 105, 114, 117] [45, 52, 59, 106, 110, 118, 122] [39, 46, 53, 97, 107, 109, 119] [54, 60, 64, 108, 111, 120, 123] [48, 55, 61, 109, 119, 121, 129] [42, 49, 56, 106, 110, 118, 122] [57, 62, 65, 108, 111, 120, 123] [51, 58, 63, 112, 116, 124, 127] [45, 52, 59, 103, 113, 115, 125] [54, 60, 64, 115, 125, 126, 131] [48, 55, 61, 112, 116, 124, 127] [57, 62, 65, 118, 122, 128, 130] [51, 58, 63, 109, 119, 121, 129] [54, 60, 64, 118, 122, 128, 130] [57, 62, 65, 115, 125, 126, 131]
H_Z (66 checks, sparse supports)
[0, 1, 6, 9, 66, 69, 74] [0, 1, 6, 9, 67, 71, 77] [2, 4, 12, 15, 68, 73, 80] [3, 11, 13, 23, 66, 69, 74] [2, 4, 12, 15, 70, 76, 83] [5, 8, 16, 20, 67, 71, 77] [6, 9, 18, 21, 72, 79, 86] [7, 17, 19, 29, 68, 73, 80] [5, 8, 16, 20, 66, 69, 74] [6, 9, 18, 21, 75, 82, 89] [10, 14, 22, 26, 70, 76, 83] [3, 11, 13, 23, 67, 71, 77] [12, 15, 24, 27, 78, 85, 92] [13, 23, 25, 35, 72, 79, 86] [10, 14, 22, 26, 68, 73, 80] [12, 15, 24, 27, 81, 88, 95] [16, 20, 28, 32, 75, 82, 89] [7, 17, 19, 29, 70, 76, 83] [18, 21, 30, 33, 84, 91, 98] [19, 29, 31, 41, 78, 85, 92] [16, 20, 28, 32, 72, 79, 86] [18, 21, 30, 33, 87, 94, 101] [22, 26, 34, 38, 81, 88, 95] [13, 23, 25, 35, 75, 82, 89] [24, 27, 36, 39, 90, 97, 104] [25, 35, 37, 47, 84, 91, 98] [22, 26, 34, 38, 78, 85, 92] [24, 27, 36, 39, 93, 100, 107] [28, 32, 40, 44, 87, 94, 101] [19, 29, 31, 41, 81, 88, 95] [30, 33, 42, 45, 96, 103, 110] [31, 41, 43, 53, 90, 97, 104] [28, 32, 40, 44, 84, 91, 98] [30, 33, 42, 45, 99, 106, 113] [34, 38, 46, 50, 93, 100, 107] [25, 35, 37, 47, 87, 94, 101] [36, 39, 48, 51, 102, 109, 116] [37, 47, 49, 59, 96, 103, 110] [34, 38, 46, 50, 90, 97, 104] [36, 39, 48, 51, 105, 112, 119] [40, 44, 52, 56, 99, 106, 113] [31, 41, 43, 53, 93, 100, 107] [42, 45, 54, 57, 108, 115, 122] [43, 53, 55, 63, 102, 109, 116] [40, 44, 52, 56, 96, 103, 110] [42, 45, 54, 57, 111, 118, 125] [46, 50, 58, 61, 105, 112, 119] [37, 47, 49, 59, 99, 106, 113] [0, 1, 48, 51, 114, 121, 127] [49, 59, 60, 65, 108, 115, 122] [46, 50, 58, 61, 102, 109, 116] [0, 1, 48, 51, 117, 124, 129] [52, 56, 62, 64, 111, 118, 125] [43, 53, 55, 63, 105, 112, 119] [2, 4, 54, 57, 120, 126, 130] [3, 11, 55, 63, 114, 121, 127] [52, 56, 62, 64, 108, 115, 122] [2, 4, 54, 57, 123, 128, 131] [5, 8, 58, 61, 117, 124, 129] [49, 59, 60, 65, 111, 118, 125] [7, 17, 60, 65, 120, 126, 130] [5, 8, 58, 61, 114, 121, 127] [10, 14, 62, 64, 123, 128, 131] [3, 11, 55, 63, 117, 124, 129] [10, 14, 62, 64, 120, 126, 130] [7, 17, 60, 65, 123, 128, 131]
Code ID 132-48-2 · download JSON · raw on GitHub