[[140,8,14]] · QEC Challenge

[[140,8,14]] d ≤

140

kd²/n

11.2

X/Z

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · 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 14 · witness weight 14 (claimed upper_bound)

witness operator (support, 14 qubits)

[9, 37, 44, 52, 74, 77, 82, 88, 89, 96, 103, 112, 132, 134]

d_Z 14 · witness weight 14 (claimed upper_bound)

witness operator (support, 14 qubits)

[9, 10, 11, 23, 32, 40, 47, 50, 51, 66, 75, 93, 110, 131]

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 7 · H_Z 7

qubit degrees H_X 2–5 (mean 3.5) · H_Z 2–5 (mean 3.5)

trapping sets H_X (1,2)×70 (2,2)×70 (3,2)×70 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 70 (1,5): 70 (2,2): 70 (2,5): 700 (2,8): 700 (3,2): 70 (3,5): 3500 (3,7): 350 (3,8): 12950 (3,9): 1960 (3,10): 1400 (3,11): 5320 (3,13): 700

trapping sets H_Z (1,2)×70 (2,2)×70 (3,2)×70 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,2): 70 (1,5): 70 (2,2): 70 (2,5): 700 (2,8): 700 (3,2): 70 (3,5): 3500 (3,7): 350 (3,8): 12950 (3,9): 1960 (3,10): 1400 (3,11): 5320 (3,13): 700

Construction & provenance

authors @MathysRennela

provenance submitted through the challenge

novelty novelty not audited

construction contributed via qldpc submit

model DeepSeek V4 Flash 0731 (claimed, not verified)

date 2026-09-18

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

[[140,8,14]] — two-block group-algebra code on SmallGroup(70,1)

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 = 11.20.

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_wt7wtL2_order70_k8.txt, line 8. This row is SmallGroup(70,1) with nonidentity GAP supports a=[5], b=[4, 23, 24, 45]. 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 ≤ 14, not an exact certificate. The published distance (d=14) 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

Union Alpha (maker currently anonymous), 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(70,1), 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] and b=[4, 23, 24, 45] 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 = 140 - 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 70 (max weight 7) · Z-checks 70 (max weight 7)

H_X (70 checks, sparse supports)

[0, 45, 70, 94, 111, 128, 129] [1, 40, 71, 99, 116, 131, 132] [1, 2, 72, 78, 104, 120, 134] [3, 54, 70, 73, 121, 134, 135] [0, 4, 74, 81, 109, 124, 136] [5, 50, 71, 75, 125, 136, 137] [4, 6, 73, 76, 84, 85, 114] [5, 7, 72, 77, 85, 128, 138] [8, 61, 73, 78, 100, 129, 138] [2, 9, 75, 79, 88, 89, 119] [3, 10, 74, 80, 89, 131, 139] [11, 58, 75, 81, 105, 132, 139] [9, 12, 77, 82, 92, 93, 123] [10, 13, 76, 78, 83, 93, 94] [11, 14, 77, 84, 94, 110, 134] [15, 66, 78, 85, 110, 111, 135] [6, 16, 80, 86, 97, 98, 127] [7, 17, 79, 81, 87, 98, 99] [8, 18, 80, 88, 99, 115, 136] [19, 64, 81, 89, 115, 116, 137] [16, 20, 83, 90, 102, 103, 130] [17, 21, 82, 84, 91, 103, 104] [18, 22, 70, 83, 85, 92, 104] [19, 23, 70, 84, 93, 120, 138] [24, 69, 85, 94, 100, 120, 121] [12, 25, 87, 95, 107, 108, 133] [13, 26, 86, 88, 96, 108, 109] [14, 27, 71, 87, 89, 97, 109] [15, 28, 71, 88, 98, 124, 139] [29, 68, 89, 99, 105, 124, 125] [25, 30, 91, 100, 112, 113, 135] [26, 31, 90, 92, 101, 113, 114] [27, 32, 72, 91, 93, 102, 114] [28, 33, 72, 73, 92, 94, 103] [29, 34, 73, 93, 104, 110, 128] [20, 35, 96, 105, 117, 118, 137] [21, 36, 95, 97, 106, 118, 119] [22, 37, 74, 96, 98, 107, 119] [23, 38, 74, 75, 97, 99, 108] [24, 39, 75, 98, 109, 115, 131] [35, 40, 101, 110, 121, 122, 138] [36, 41, 100, 102, 111, 122, 123] [37, 42, 76, 101, 103, 112, 123] [38, 43, 76, 77, 102, 104, 113] [39, 44, 70, 77, 78, 103, 114] [30, 45, 106, 115, 125, 126, 139] [31, 46, 105, 107, 116, 126, 127] [32, 47, 79, 106, 108, 117, 127] [33, 48, 79, 80, 107, 109, 118] [34, 49, 71, 80, 81, 108, 119] [46, 50, 110, 112, 120, 129, 130] [47, 51, 82, 111, 113, 121, 130] [48, 52, 82, 83, 112, 114, 122] [49, 53, 72, 83, 84, 113, 123] [41, 54, 115, 117, 124, 132, 133] [42, 55, 86, 116, 118, 125, 133] [43, 56, 86, 87, 117, 119, 126] [44, 57, 74, 87, 88, 118, 127] [55, 58, 90, 120, 122, 128, 135] [56, 59, 90, 91, 121, 123, 129] [57, 60, 76, 91, 92, 122, 130] [51, 61, 95, 124, 126, 131, 137] [52, 62, 95, 96, 125, 127, 132] [53, 63, 79, 96, 97, 126, 133] [62, 64, 100, 101, 128, 130, 134] [63, 65, 82, 101, 102, 129, 135] [59, 66, 105, 106, 131, 133, 136] [60, 67, 86, 106, 107, 132, 137] [67, 68, 90, 111, 112, 134, 138] [65, 69, 95, 116, 117, 136, 139]

H_Z (70 checks, sparse supports)

[0, 3, 22, 23, 44, 70, 74] [1, 5, 27, 28, 49, 71, 72] [2, 7, 32, 33, 53, 72, 79] [3, 6, 8, 33, 34, 73, 80] [4, 10, 37, 38, 57, 74, 76] [5, 9, 11, 38, 39, 75, 77] [6, 13, 42, 43, 60, 76, 86] [7, 12, 14, 43, 44, 77, 87] [2, 8, 13, 15, 44, 78, 88] [9, 17, 47, 48, 63, 79, 82] [10, 16, 18, 48, 49, 80, 83] [4, 11, 17, 19, 49, 81, 84] [12, 21, 51, 52, 65, 82, 95] [13, 20, 22, 52, 53, 83, 96] [6, 14, 21, 23, 53, 84, 97] [6, 7, 15, 22, 24, 85, 98] [16, 26, 55, 56, 67, 86, 90] [17, 25, 27, 56, 57, 87, 91] [9, 18, 26, 28, 57, 88, 92] [9, 10, 19, 27, 29, 89, 93] [20, 31, 58, 59, 68, 90, 105] [21, 30, 32, 59, 60, 91, 106] [12, 22, 31, 33, 60, 92, 107] [12, 13, 23, 32, 34, 93, 108] [0, 13, 14, 24, 33, 94, 109] [25, 36, 61, 62, 69, 95, 100] [26, 35, 37, 62, 63, 96, 101] [16, 27, 36, 38, 63, 97, 102] [16, 17, 28, 37, 39, 98, 103] [1, 17, 18, 29, 38, 99, 104] [8, 24, 30, 41, 64, 100, 115] [31, 40, 42, 64, 65, 101, 116] [20, 32, 41, 43, 65, 102, 117] [20, 21, 33, 42, 44, 103, 118] [2, 21, 22, 34, 43, 104, 119] [11, 29, 35, 46, 66, 105, 110] [36, 45, 47, 66, 67, 106, 111] [25, 37, 46, 48, 67, 107, 112] [25, 26, 38, 47, 49, 108, 113] [4, 26, 27, 39, 48, 109, 114] [14, 15, 34, 40, 50, 71, 110] [0, 15, 41, 51, 68, 111, 124] [30, 42, 50, 52, 68, 112, 125] [30, 31, 43, 51, 53, 113, 126] [6, 31, 32, 44, 52, 114, 127] [18, 19, 39, 45, 54, 70, 115] [1, 19, 46, 55, 69, 116, 120] [35, 47, 54, 56, 69, 117, 121] [35, 36, 48, 55, 57, 118, 122] [9, 36, 37, 49, 56, 119, 123] [2, 23, 24, 50, 58, 75, 120] [3, 24, 40, 51, 59, 121, 131] [40, 41, 52, 58, 60, 122, 132] [12, 41, 42, 53, 59, 123, 133] [4, 28, 29, 54, 61, 73, 124] [5, 29, 45, 55, 62, 125, 128] [45, 46, 56, 61, 63, 126, 129] [16, 46, 47, 57, 62, 127, 130] [0, 7, 34, 58, 64, 81, 128] [0, 8, 50, 59, 65, 129, 136] [20, 50, 51, 60, 64, 130, 137] [1, 10, 39, 61, 66, 78, 131] [1, 11, 54, 62, 67, 132, 134] [25, 54, 55, 63, 66, 133, 135] [2, 3, 14, 64, 68, 89, 134] [3, 15, 30, 58, 65, 135, 139] [4, 5, 18, 66, 69, 85, 136] [5, 19, 35, 61, 67, 137, 138] [7, 8, 23, 40, 68, 99, 138] [10, 11, 28, 45, 69, 94, 139]

Code ID 140-8-14 · download JSON · raw on GitHub