[[136,4,18]] · QEC Challenge

[[136,4,18]] d ≤

136

kd²/n

9.529

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 18, d_Z ≤ 18 · 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 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[3, 7, 18, 22, 34, 37, 50, 53, 66, 91, 93, 97, 106, 111, 112, 116, 119, 120]

d_Z 18 · witness weight 18 (claimed upper_bound)

witness operator (support, 18 qubits)

[0, 12, 14, 17, 43, 52, 65, 66, 68, 81, 89, 95, 104, 108, 109, 116, 120, 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 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 4 · H_Z 4

trapping sets H_X (1,4)×136 (2,6)×1904 (3,6)×1836 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 136 (2,6): 1904 (3,6): 1836 (3,8): 34476 (3,10): 3808

trapping sets H_Z (1,4)×136 (2,6)×1904 (3,6)×1836 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,4): 136 (2,6): 1904 (3,6): 1836 (3,8): 34476 (3,10): 3808

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_wt8wtL4_order68_k4.txt, line 11. nonidentity supports a=[2, 4, 13], b=[15, 28, 44].

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

[[136,4,18]] — 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 = 9.53.

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_wt8wtL4_order68_k4.txt, line 11. This row is SmallGroup(68,1) with nonidentity GAP supports a=[2, 4, 13], b=[15, 28, 44]. 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=4), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 18, 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=[2, 4, 13], b=[15, 28, 44] 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) = 4. 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, 4, 9, 63, 68, 91, 107, 126] [0, 1, 5, 62, 69, 93, 109, 128] [1, 2, 12, 66, 70, 94, 110, 123] [0, 3, 8, 13, 71, 95, 111, 130] [2, 4, 8, 59, 72, 96, 112, 125] [3, 5, 9, 66, 73, 97, 113, 132] [2, 5, 6, 16, 74, 98, 114, 127] [3, 7, 12, 17, 75, 99, 115, 134] [6, 8, 12, 63, 76, 100, 116, 129] [2, 7, 9, 13, 77, 101, 117, 135] [6, 9, 10, 20, 78, 102, 118, 131] [7, 11, 16, 21, 70, 79, 103, 119] [0, 10, 12, 16, 80, 104, 120, 133] [6, 11, 13, 17, 72, 81, 105, 121] [10, 13, 14, 24, 68, 82, 106, 122] [11, 15, 20, 25, 74, 83, 107, 123] [3, 14, 16, 20, 69, 84, 108, 124] [10, 15, 17, 21, 76, 85, 109, 125] [14, 17, 18, 28, 71, 86, 110, 126] [15, 19, 24, 29, 78, 87, 111, 127] [7, 18, 20, 24, 73, 88, 112, 128] [14, 19, 21, 25, 80, 89, 113, 129] [18, 21, 22, 32, 75, 90, 114, 130] [19, 23, 28, 33, 82, 91, 115, 131] [11, 22, 24, 28, 77, 92, 116, 132] [18, 23, 25, 29, 84, 93, 117, 133] [22, 25, 26, 36, 79, 94, 118, 134] [23, 27, 32, 37, 68, 86, 95, 119] [15, 26, 28, 32, 81, 96, 120, 135] [22, 27, 29, 33, 69, 88, 97, 121] [26, 29, 30, 40, 70, 83, 98, 122] [27, 31, 36, 41, 71, 90, 99, 123] [19, 30, 32, 36, 72, 85, 100, 124] [26, 31, 33, 37, 73, 92, 101, 125] [30, 33, 34, 44, 74, 87, 102, 126] [31, 35, 40, 45, 75, 94, 103, 127] [23, 34, 36, 40, 76, 89, 104, 128] [30, 35, 37, 41, 77, 96, 105, 129] [34, 37, 38, 48, 78, 91, 106, 130] [35, 39, 44, 49, 79, 98, 107, 131] [27, 38, 40, 44, 80, 93, 108, 132] [34, 39, 41, 45, 81, 100, 109, 133] [38, 41, 42, 52, 82, 95, 110, 134] [39, 43, 48, 53, 68, 83, 102, 111] [31, 42, 44, 48, 84, 97, 112, 135] [38, 43, 45, 49, 69, 85, 104, 113] [42, 45, 46, 56, 70, 86, 99, 114] [43, 47, 52, 57, 71, 87, 106, 115] [35, 46, 48, 52, 72, 88, 101, 116] [42, 47, 49, 53, 73, 89, 108, 117] [46, 49, 50, 60, 74, 90, 103, 118] [47, 51, 56, 61, 75, 91, 110, 119] [39, 50, 52, 56, 76, 92, 105, 120] [46, 51, 53, 57, 77, 93, 112, 121] [50, 53, 54, 64, 78, 94, 107, 122] [51, 55, 60, 65, 79, 95, 114, 123] [43, 54, 56, 60, 80, 96, 109, 124] [50, 55, 57, 61, 81, 97, 116, 125] [54, 57, 58, 67, 82, 98, 111, 126] [1, 55, 59, 64, 83, 99, 118, 127] [47, 58, 60, 64, 84, 100, 113, 128] [54, 59, 61, 65, 85, 101, 120, 129] [4, 58, 61, 62, 86, 102, 115, 130] [5, 59, 63, 67, 87, 103, 122, 131] [51, 62, 64, 67, 88, 104, 117, 132] [1, 58, 63, 65, 89, 105, 124, 133] [8, 62, 65, 66, 90, 106, 119, 134] [4, 55, 66, 67, 92, 108, 121, 135]

H_Z (68 checks, sparse supports)

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

Code ID 136-4-18 · download JSON · raw on GitHub