[[136,8,15]] · QEC Challenge

[[136,8,15]] d ≤

136

kd²/n

13.235

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (claimed upper_bound)

witness operator (support, 15 qubits)

[1, 9, 17, 25, 33, 41, 49, 57, 79, 100, 108, 116, 117, 124, 125]

d_Z 15 · witness weight 15 (claimed upper_bound)

witness operator (support, 15 qubits)

[0, 2, 12, 16, 22, 27, 31, 54, 59, 76, 77, 85, 94, 103, 119]

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 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 (2,6): 816 (2,10): 1020 (3,2): 68 (3,4): 68 (3,6): 4692 (3,8): 1088 (3,10): 20808 (3,12): 5372 (3,14): 15504 (3,16): 1360

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 (2,6): 816 (2,10): 1020 (3,2): 68 (3,4): 68 (3,6): 4692 (3,8): 1088 (3,10): 20808 (3,12): 5372 (3,14): 15504 (3,16): 1360

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_k8.txt, line 11. nonidentity supports a=[4], b=[2, 7, 10, 16, 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

[[136,8,15]] — 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 = 13.24.

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_k8.txt, line 11. This row is SmallGroup(68,1) with nonidentity GAP supports a=[4], b=[2, 7, 10, 16, 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=8), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 15, 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, 7, 10, 16, 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) = 8. 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, 80, 102, 119, 134] [1, 5, 68, 69, 75, 104, 121, 135] [2, 66, 69, 70, 77, 99, 122, 131] [0, 3, 70, 71, 76, 106, 123, 135] [4, 8, 70, 72, 78, 101, 124, 133] [5, 9, 71, 72, 73, 108, 125, 131] [2, 6, 68, 73, 74, 103, 126, 133] [3, 7, 72, 74, 75, 110, 127, 132] [8, 12, 69, 74, 76, 105, 128, 134] [9, 13, 68, 76, 77, 112, 127, 129] [6, 10, 69, 71, 78, 107, 129, 130] [7, 11, 78, 79, 114, 128, 131, 135] [12, 16, 70, 73, 80, 109, 130, 132] [13, 17, 80, 81, 116, 123, 131, 133] [10, 14, 75, 82, 111, 125, 133, 134] [11, 15, 68, 82, 83, 118, 124, 132] [16, 20, 77, 84, 113, 126, 134, 135] [17, 21, 69, 84, 85, 119, 120, 127] [14, 18, 70, 79, 86, 115, 121, 129] [15, 19, 71, 86, 87, 120, 122, 128] [20, 24, 72, 81, 88, 117, 122, 130] [21, 25, 73, 88, 89, 115, 123, 124] [18, 22, 74, 83, 90, 117, 119, 125] [19, 23, 75, 90, 91, 116, 124, 126] [24, 28, 76, 85, 92, 118, 121, 126] [25, 29, 77, 92, 93, 111, 119, 128] [22, 26, 78, 87, 94, 113, 121, 123] [23, 27, 79, 94, 95, 112, 120, 130] [28, 32, 80, 89, 96, 114, 122, 125] [29, 33, 81, 96, 97, 107, 115, 132] [26, 30, 82, 91, 98, 109, 117, 127] [27, 31, 83, 98, 99, 108, 116, 134] [32, 36, 84, 93, 100, 110, 118, 129] [33, 37, 85, 100, 101, 103, 111, 135] [30, 34, 86, 95, 102, 105, 113, 131] [31, 35, 70, 87, 102, 103, 104, 112] [36, 40, 88, 97, 104, 106, 114, 133] [37, 41, 72, 89, 99, 104, 105, 107] [34, 38, 68, 90, 99, 101, 106, 109] [35, 39, 74, 91, 100, 106, 107, 108] [40, 44, 69, 92, 101, 102, 108, 110] [41, 45, 76, 93, 95, 103, 108, 109] [38, 42, 71, 94, 97, 103, 105, 110] [39, 43, 78, 95, 96, 104, 110, 111] [44, 48, 73, 96, 98, 105, 106, 112] [45, 49, 80, 91, 97, 99, 112, 113] [42, 46, 75, 93, 98, 101, 107, 114] [43, 47, 82, 92, 99, 100, 114, 115] [48, 52, 77, 94, 100, 102, 109, 116] [49, 53, 84, 87, 95, 101, 116, 117] [46, 50, 79, 89, 97, 102, 111, 118] [47, 51, 86, 88, 96, 103, 118, 119] [52, 56, 81, 90, 98, 104, 113, 120] [53, 57, 83, 88, 91, 105, 120, 121] [50, 54, 83, 85, 93, 106, 115, 122] [51, 55, 84, 90, 92, 107, 122, 123] [56, 60, 85, 86, 94, 108, 117, 124] [57, 61, 79, 87, 92, 109, 124, 125] [54, 58, 81, 87, 89, 110, 119, 126] [55, 59, 80, 88, 94, 111, 126, 127] [60, 64, 82, 89, 90, 112, 121, 128] [61, 65, 75, 83, 96, 113, 128, 129] [58, 62, 77, 85, 91, 114, 123, 130] [59, 63, 76, 84, 98, 115, 130, 131] [64, 67, 78, 86, 93, 116, 125, 132] [1, 65, 71, 79, 100, 117, 132, 133] [62, 66, 73, 81, 95, 118, 127, 134] [4, 67, 74, 82, 97, 120, 129, 135]

H_Z (68 checks, sparse supports)

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

Code ID 136-8-15 · download JSON · raw on GitHub