[[136,6,16]] · QEC Challenge

[[136,6,16]] d ≤

136

kd²/n

11.294

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · 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 16 · witness weight 16 (claimed upper_bound)

witness operator (support, 16 qubits)

[3, 44, 71, 72, 73, 76, 77, 80, 81, 84, 85, 88, 102, 106, 108, 135]

d_Z 16 · witness weight 16 (claimed upper_bound)

witness operator (support, 16 qubits)

[12, 15, 16, 17, 19, 33, 34, 37, 38, 50, 52, 64, 85, 116, 124, 132]

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)×68 (2,2)×68 (3,2)×68 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 68 (1,6): 68 (2,2): 68 (2,4): 68 (2,6): 680 (2,8): 68 (2,10): 884 (3,2): 68 (3,4): 544 (3,6): 3604 (3,8): 2992 (3,10): 16252 (3,12): 4964 (3,14): 13328 (3,16): 884

trapping sets H_Z (1,2)×68 (2,2)×68 (3,2)×68 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 68 (1,6): 68 (2,2): 68 (2,4): 68 (2,6): 680 (2,8): 68 (2,10): 884 (3,2): 68 (3,4): 544 (3,6): 3604 (3,8): 2992 (3,10): 16252 (3,12): 4964 (3,14): 13328 (3,16): 884

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction 2BGA (arXiv:2306.16400) on SmallGroup(68,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order68_k6.txt, line 9. nonidentity supports a=[4], b=[2, 4, 20, 29, 39].

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

[[136,6,16]] — two-block group-algebra code on SmallGroup(68,1)

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 = 11.29.

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_order68_k6.txt, line 9. This row is SmallGroup(68,1) with nonidentity GAP supports a=[4], b=[2, 4, 20, 29, 39]. 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=6), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 16, 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(68,1), 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], b=[2, 4, 20, 29, 39] 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 = 136 - rank(HX) - rank(HZ) = 6. 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 68 (max weight 8) · Z-checks 68 (max weight 8)

H_X (68 checks, sparse supports)

[0, 63, 68, 72, 93, 102, 115, 131] [1, 5, 68, 69, 94, 104, 117, 133] [2, 66, 69, 70, 96, 99, 118, 134] [0, 3, 68, 71, 89, 106, 119, 135] [4, 8, 70, 72, 91, 101, 120, 135] [5, 9, 69, 73, 90, 108, 121, 131] [2, 6, 70, 74, 92, 103, 122, 133] [3, 7, 71, 75, 85, 110, 123, 132] [8, 12, 72, 76, 87, 105, 124, 134] [9, 13, 73, 77, 86, 112, 125, 127] [6, 10, 74, 78, 88, 107, 126, 129] [7, 11, 75, 79, 81, 114, 127, 128] [12, 16, 76, 80, 83, 109, 128, 130] [13, 17, 77, 81, 82, 116, 123, 129] [10, 14, 78, 82, 84, 111, 125, 130] [11, 15, 77, 79, 83, 118, 124, 131] [16, 20, 79, 80, 84, 113, 126, 132] [17, 21, 78, 81, 85, 119, 120, 133] [14, 18, 80, 82, 86, 115, 121, 134] [15, 19, 68, 73, 83, 87, 120, 122] [20, 24, 75, 84, 88, 117, 122, 135] [21, 25, 69, 74, 85, 89, 115, 124] [18, 22, 70, 76, 86, 90, 117, 119] [19, 23, 69, 71, 87, 91, 116, 126] [24, 28, 71, 72, 88, 92, 118, 121] [25, 29, 70, 73, 89, 93, 111, 128] [22, 26, 72, 74, 90, 94, 113, 123] [23, 27, 75, 91, 95, 112, 130, 133] [28, 32, 68, 76, 92, 96, 114, 125] [29, 33, 77, 93, 97, 107, 132, 134] [26, 30, 78, 94, 98, 109, 127, 135] [27, 31, 79, 95, 99, 108, 129, 134] [32, 36, 80, 96, 100, 110, 129, 131] [33, 37, 81, 97, 101, 103, 130, 135] [30, 34, 82, 98, 102, 105, 131, 132] [31, 35, 70, 83, 99, 103, 104, 125] [36, 40, 84, 100, 104, 106, 127, 133] [37, 41, 72, 85, 99, 101, 105, 126] [34, 38, 68, 86, 101, 102, 106, 128] [35, 39, 74, 87, 100, 103, 107, 121] [40, 44, 69, 88, 102, 104, 108, 123] [41, 45, 76, 89, 95, 105, 109, 122] [38, 42, 71, 90, 97, 106, 110, 124] [39, 43, 78, 91, 96, 107, 111, 117] [44, 48, 73, 92, 98, 108, 112, 119] [45, 49, 80, 91, 93, 109, 113, 118] [42, 46, 75, 93, 94, 110, 114, 120] [43, 47, 82, 92, 95, 111, 113, 115] [48, 52, 77, 94, 96, 112, 115, 116] [49, 53, 84, 87, 97, 113, 114, 117] [46, 50, 79, 89, 98, 114, 116, 118] [47, 51, 86, 88, 99, 109, 115, 119] [52, 56, 81, 90, 100, 111, 116, 120] [53, 57, 83, 88, 101, 110, 117, 121] [50, 54, 83, 85, 102, 112, 118, 122] [51, 55, 84, 90, 103, 105, 119, 123] [56, 60, 85, 86, 104, 107, 120, 124] [57, 61, 79, 92, 105, 106, 121, 125] [54, 58, 81, 87, 106, 108, 122, 126] [55, 59, 80, 94, 101, 107, 123, 127] [60, 64, 82, 89, 103, 108, 124, 128] [61, 65, 75, 96, 102, 109, 125, 129] [58, 62, 77, 91, 104, 110, 126, 130] [59, 63, 76, 97, 98, 111, 127, 131] [64, 67, 78, 93, 99, 112, 128, 132] [1, 65, 71, 98, 100, 113, 129, 133] [62, 66, 73, 95, 100, 114, 130, 134] [4, 67, 74, 95, 97, 116, 132, 135]

H_Z (68 checks, sparse supports)

[0, 1, 3, 19, 28, 38, 68, 71] [1, 2, 5, 21, 23, 40, 69, 133] [2, 4, 6, 22, 25, 35, 70, 74] [3, 7, 23, 24, 42, 65, 71, 75] [0, 4, 8, 24, 26, 37, 72, 135] [5, 9, 19, 25, 44, 66, 69, 73] [6, 10, 21, 26, 39, 67, 74, 78] [7, 11, 20, 27, 46, 61, 75, 79] [8, 12, 22, 28, 41, 63, 72, 76] [9, 13, 15, 29, 48, 62, 73, 77] [10, 14, 17, 30, 43, 64, 78, 82] [11, 15, 16, 31, 50, 57, 79, 83] [12, 16, 18, 32, 45, 59, 76, 80] [11, 13, 17, 33, 52, 58, 77, 81] [13, 14, 18, 34, 47, 60, 82, 86] [12, 15, 19, 35, 53, 54, 83, 87] [14, 16, 20, 36, 49, 55, 80, 84] [7, 17, 21, 37, 54, 56, 81, 85] [9, 18, 22, 38, 51, 56, 86, 90] [8, 19, 23, 39, 49, 58, 87, 91] [10, 20, 24, 40, 51, 53, 84, 88] [3, 21, 25, 41, 50, 60, 85, 89] [5, 22, 26, 42, 52, 55, 90, 94] [4, 23, 27, 43, 45, 62, 91, 95] [6, 24, 28, 44, 47, 57, 88, 92] [0, 25, 29, 45, 46, 64, 89, 93] [1, 26, 30, 46, 48, 59, 94, 98] [27, 31, 41, 47, 66, 67, 95, 99] [2, 28, 32, 43, 48, 61, 92, 96] [29, 33, 42, 49, 63, 67, 93, 97] [30, 34, 44, 50, 63, 65, 98, 102] [2, 31, 35, 37, 51, 64, 99, 103] [32, 36, 39, 52, 65, 66, 96, 100] [4, 33, 37, 38, 53, 59, 97, 101] [0, 34, 38, 40, 54, 61, 102, 106] [6, 33, 35, 39, 55, 60, 103, 107] [1, 35, 36, 40, 56, 62, 100, 104] [8, 34, 37, 41, 55, 57, 101, 105] [3, 36, 38, 42, 57, 58, 106, 110] [10, 29, 39, 43, 56, 59, 107, 111] [5, 31, 40, 44, 58, 60, 104, 108] [12, 30, 41, 45, 51, 61, 105, 109] [7, 32, 42, 46, 53, 62, 110, 114] [14, 25, 43, 47, 52, 63, 111, 115] [9, 27, 44, 48, 54, 64, 108, 112] [16, 26, 45, 47, 49, 65, 109, 113] [11, 28, 46, 49, 50, 66, 114, 118] [0, 18, 21, 47, 48, 51, 115, 119] [13, 23, 48, 50, 52, 67, 112, 116] [1, 20, 22, 43, 49, 53, 113, 117] [2, 15, 24, 45, 50, 54, 118, 122] [3, 17, 22, 44, 51, 55, 119, 123] [4, 17, 19, 46, 52, 56, 116, 120] [5, 18, 24, 39, 53, 57, 117, 121] [6, 19, 20, 41, 54, 58, 122, 126] [7, 13, 26, 40, 55, 59, 123, 127] [8, 15, 21, 42, 56, 60, 120, 124] [9, 14, 28, 35, 57, 61, 121, 125] [10, 16, 23, 37, 58, 62, 126, 130] [9, 11, 30, 36, 59, 63, 127, 131] [11, 12, 25, 38, 60, 64, 124, 128] [10, 13, 31, 32, 61, 65, 125, 129] [12, 14, 27, 33, 62, 66, 130, 134] [0, 5, 15, 32, 34, 63, 68, 131] [7, 16, 29, 34, 64, 67, 128, 132] [1, 6, 17, 27, 36, 65, 129, 133] [2, 8, 18, 29, 31, 66, 70, 134] [3, 4, 20, 30, 33, 67, 132, 135]

Code ID 136-6-16 · download JSON · raw on GitHub