[[132,2,17]] · QEC Challenge

[[132,2,17]] d ≤

132

kd²/n

4.379

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · 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 17 · witness weight 17 (claimed upper_bound)

witness operator (support, 17 qubits)

[8, 27, 29, 46, 63, 79, 94, 99, 102, 105, 108, 109, 111, 114, 117, 125, 130]

d_Z 17 · witness weight 17 (claimed upper_bound)

witness operator (support, 17 qubits)

[8, 12, 15, 23, 26, 37, 41, 49, 50, 51, 74, 97, 102, 105, 111, 115, 128]

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 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 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)

trapping sets H_X (1,2)×66 (2,2)×66 (3,2)×66 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_X

(1,2): 66 (1,6): 66 (2,2): 66 (2,4): 22 (2,6): 748 (2,8): 33 (2,10): 924 (3,2): 66 (3,4): 176 (3,6): 4356 (3,8): 2068 (3,10): 17688 (3,12): 5082 (3,14): 13596 (3,16): 1056

trapping sets H_Z (1,2)×66 (2,2)×66 (3,2)×66 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

full (size, syndrome weight): count census for H_Z

(1,2): 66 (1,6): 66 (2,2): 66 (2,4): 22 (2,6): 748 (2,8): 33 (2,10): 924 (3,2): 66 (3,4): 176 (3,6): 4356 (3,8): 2068 (3,10): 17688 (3,12): 5082 (3,14): 13596 (3,16): 1056

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_wt8wtL2_order66_k2.txt, line 8. nonidentity supports a=[5], b=[2, 3, 14, 39, 40].

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

[[132,2,17]] — two-block group-algebra code on SmallGroup(66,1)

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 = 4.38.

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_wt8wtL2_order66_k2.txt, line 8. This row is SmallGroup(66,1) with nonidentity GAP supports a=[5], b=[2, 3, 14, 39, 40]. 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=2), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 17, 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(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=[5], b=[2, 3, 14, 39, 40] 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) = 2. 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 8) · Z-checks 66 (max weight 8)

H_X (66 checks, sparse supports)

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

H_Z (66 checks, sparse supports)

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

Code ID 132-2-17 · download JSON · raw on GitHub