[[140,6,15]] · QEC Challenge

[[140,6,15]] d ≤

140

kd²/n

9.643

X/Z

Share this result

Distance

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

witness operator (support, 15 qubits)

[11, 18, 65, 76, 84, 93, 95, 96, 101, 103, 106, 111, 113, 128, 138]

d_Z 15 · witness weight 15 (claimed upper_bound)

witness operator (support, 15 qubits)

[10, 17, 22, 25, 26, 29, 35, 43, 52, 67, 92, 94, 97, 101, 105]

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 7 · H_Z 7

qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)

trapping sets H_X (1,3)×70 (2,3)×35 (3,3)×70 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 70 (1,4): 70 (2,3): 35 (2,4): 210 (2,5): 770 (2,6): 420 (3,3): 70 (3,4): 420 (3,5): 1330 (3,6): 6195 (3,7): 8995 (3,8): 3850 (3,9): 1050 (3,10): 280

trapping sets H_Z (1,3)×70 (2,3)×35 (3,3)×70 (smallest syndrome weight at each size, connected sets of up to 3 qubits)

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

(1,3): 70 (1,4): 70 (2,3): 35 (2,4): 210 (2,5): 770 (2,6): 420 (3,3): 70 (3,4): 420 (3,5): 1330 (3,6): 6195 (3,7): 8995 (3,8): 3850 (3,9): 1050 (3,10): 280

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

[[140,6,15]] — 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 = 9.64.

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_wt7wtL3_order70_k6.txt, line 6. This row is SmallGroup(70,1) with nonidentity GAP supports a=[3, 22], b=[5, 43, 51]. 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 ≤ 15, not an exact certificate. The published distance (d=15) 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=[3, 22] and b=[5, 43, 51] 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) = 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 70 (max weight 7) · Z-checks 70 (max weight 7)

H_X (70 checks, sparse supports)

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

H_Z (70 checks, sparse supports)

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

Code ID 140-6-15 · download JSON · raw on GitHub