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[[140,20,6]] d ≤
n
140
k
20
d
6
kd²/n
5.143
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 6, d_Z ≤ 6 · w_X = 6, w_Z = 6 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[45, 61, 69, 121, 135, 139]
d_Z 6 · witness weight 6 (claimed upper_bound)
witness operator (support, 6 qubits)
[42, 52, 55, 62, 67, 112]
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 6 · H_Z 6
qubit degrees H_X 2–4 (mean 3.0) · H_Z 2–4 (mean 3.0)
trapping sets H_X (1,2)×70 (2,2)×70 (3,2)×140 (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 (1,4): 70 (2,2): 70 (2,4): 560 (2,6): 420 (3,2): 140 (3,4): 2380 (3,6): 5880 (3,8): 3150 (3,10): 280
trapping sets H_Z (1,2)×70 (2,2)×70 (3,2)×140 (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 (1,4): 70 (2,2): 70 (2,4): 560 (2,6): 420 (3,2): 140 (3,4): 2380 (3,6): 5880 (3,8): 3150 (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 ≤ 6 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[140,20,6]] — two-block group-algebra code on SmallGroup(70,2)

Direction & hypothesis

Target: unrestricted weight-6 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 = 5.14.

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_wt6wtL2_order70_k20.txt, line 4. This row is SmallGroup(70,2) with nonidentity GAP supports a=[4], b=[5, 9, 29]. 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=20), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 6, not an exact certificate. The published distance (d=6) 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(70,2), 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] and b=[5, 9, 29] 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) = 20. 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 6) · Z-checks 70 (max weight 6)
H_X (70 checks, sparse supports)
[0, 44, 70, 95, 104, 124] [1, 5, 71, 90, 109, 120] [2, 53, 71, 72, 89, 113] [0, 3, 73, 114, 115, 139] [4, 10, 70, 74, 85, 118] [5, 11, 75, 110, 119, 138] [6, 60, 74, 76, 98, 122] [2, 7, 77, 81, 119, 123] [3, 8, 70, 78, 105, 136] [9, 17, 72, 79, 93, 126] [10, 18, 78, 80, 114, 127] [11, 19, 71, 81, 100, 134] [12, 65, 79, 82, 107, 129] [6, 13, 83, 88, 127, 130] [7, 14, 72, 75, 84, 109] [8, 15, 73, 85, 95, 131] [16, 26, 76, 86, 102, 132] [17, 27, 84, 87, 123, 133] [18, 28, 73, 74, 88, 104] [19, 29, 75, 89, 90, 128] [20, 68, 86, 90, 116, 134] [12, 21, 91, 97, 133, 135] [13, 22, 76, 80, 92, 118] [14, 23, 71, 77, 93, 99] [15, 24, 78, 94, 124, 139] [25, 35, 82, 95, 111, 136] [26, 36, 92, 96, 130, 137] [27, 37, 77, 79, 97, 113] [28, 38, 70, 80, 94, 98] [29, 39, 81, 99, 120, 138] [20, 30, 100, 106, 137, 138] [21, 31, 82, 87, 101, 126] [22, 32, 74, 83, 102, 108] [23, 33, 84, 89, 103, 119] [24, 34, 85, 104, 115, 136] [35, 45, 101, 105, 135, 139] [36, 46, 83, 86, 106, 122] [37, 47, 72, 87, 103, 107] [38, 48, 85, 88, 108, 114] [39, 49, 89, 109, 110, 134] [30, 40, 90, 96, 110, 132] [31, 41, 79, 91, 111, 117] [32, 42, 92, 98, 112, 127] [33, 43, 81, 93, 109, 113] [34, 44, 94, 105, 114, 131] [45, 54, 91, 95, 115, 129] [46, 55, 76, 96, 112, 116] [47, 56, 93, 97, 117, 123] [48, 57, 78, 98, 104, 118] [1, 49, 99, 100, 119, 128] [40, 50, 86, 100, 120, 125] [41, 51, 101, 107, 121, 133] [42, 52, 88, 102, 118, 122] [43, 53, 75, 99, 103, 123] [54, 61, 82, 105, 121, 124] [55, 62, 102, 106, 125, 130] [56, 63, 84, 107, 113, 126] [4, 57, 73, 94, 108, 127] [50, 58, 110, 116, 128, 137] [51, 59, 97, 111, 126, 129] [52, 60, 80, 108, 112, 130] [61, 66, 111, 115, 131, 135] [62, 67, 92, 116, 122, 132] [9, 63, 77, 103, 117, 133] [58, 64, 106, 120, 132, 134] [59, 65, 87, 117, 121, 135] [66, 69, 101, 124, 129, 136] [16, 67, 83, 112, 125, 137] [64, 68, 96, 125, 128, 138] [25, 69, 91, 121, 131, 139]
H_Z (70 checks, sparse supports)
[0, 4, 8, 28, 70, 73] [1, 2, 11, 23, 71, 119] [2, 9, 14, 37, 72, 77] [3, 15, 18, 57, 73, 78] [4, 6, 18, 32, 74, 127] [5, 14, 19, 53, 71, 75] [6, 16, 22, 46, 76, 83] [7, 23, 27, 63, 77, 84] [8, 10, 24, 48, 78, 85] [9, 12, 27, 41, 79, 133] [10, 22, 28, 60, 74, 80] [7, 11, 29, 43, 75, 81] [12, 25, 31, 54, 82, 91] [13, 32, 36, 67, 83, 92] [14, 17, 33, 56, 84, 93] [4, 15, 34, 38, 85, 94] [16, 20, 36, 50, 86, 137] [17, 31, 37, 65, 79, 87] [13, 18, 38, 52, 80, 88] [2, 19, 33, 39, 81, 89] [1, 19, 20, 40, 90, 100] [21, 41, 45, 69, 91, 101] [22, 26, 42, 62, 92, 102] [9, 23, 43, 47, 93, 103] [24, 28, 44, 57, 94, 104] [0, 15, 25, 45, 95, 139] [26, 40, 46, 68, 86, 96] [21, 27, 47, 59, 87, 97] [6, 28, 42, 48, 88, 98] [23, 29, 49, 53, 89, 99] [11, 30, 49, 50, 100, 110] [31, 35, 51, 66, 101, 111] [16, 32, 52, 55, 102, 112] [33, 37, 53, 63, 103, 113] [0, 18, 34, 48, 104, 114] [8, 35, 44, 54, 95, 105] [30, 36, 55, 64, 96, 106] [12, 37, 51, 56, 97, 107] [32, 38, 57, 60, 98, 108] [1, 14, 39, 43, 99, 109] [5, 39, 40, 58, 110, 120] [25, 41, 59, 61, 111, 121] [42, 46, 60, 67, 112, 122] [2, 27, 43, 56, 113, 123] [3, 10, 38, 44, 70, 114] [3, 34, 45, 61, 105, 115] [20, 46, 58, 62, 106, 116] [41, 47, 63, 65, 107, 117] [4, 22, 48, 52, 108, 118] [5, 7, 33, 49, 109, 119] [1, 29, 50, 64, 120, 128] [51, 54, 65, 69, 121, 129] [6, 36, 52, 62, 122, 130] [7, 17, 47, 53, 72, 123] [0, 24, 54, 66, 115, 124] [50, 55, 67, 68, 116, 125] [9, 31, 56, 59, 117, 126] [10, 13, 42, 57, 118, 127] [19, 49, 58, 68, 128, 134] [12, 45, 59, 66, 129, 135] [13, 26, 55, 60, 76, 130] [15, 44, 61, 69, 124, 131] [16, 40, 62, 64, 125, 132] [17, 21, 51, 63, 126, 133] [11, 20, 39, 64, 134, 138] [21, 35, 61, 65, 82, 135] [8, 25, 34, 66, 131, 136] [26, 30, 58, 67, 132, 137] [5, 29, 30, 68, 90, 138] [3, 24, 35, 69, 136, 139]
Code ID 140-20-6 · download JSON · raw on GitHub