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[[140,42,2]] d =
n
140
k
42
d
2
kd²/n
1.2
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[82, 86]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[52, 56]
certificate exact, d = 2 · scipy/HiGHS MILP
X: no logical < 2 exists; Z: no logical < 2 exists

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 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 (2,0)×70 (3,0)×56 (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 (2,0): 70 (2,2): 56 (2,4): 448 (2,6): 280 (3,0): 56 (3,2): 252 (3,4): 2100 (3,6): 3696 (3,8): 2016
trapping sets H_Z (1,2)×70 (2,0)×35 (3,2)×448 (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 (2,0): 35 (2,2): 56 (2,4): 658 (3,2): 448 (3,4): 3234 (3,6): 1596 (3,8): 140

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(70,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt6wtL2_order70_k42.txt, line 1. nonidentity supports a=[2], b=[2, 8, 11].
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 ≤ 6 (computed)

How this code was found

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

[[140,42,2]] — two-block group-algebra code on SmallGroup(70,1)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-6 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 1.20.

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_wt6wtL2_order70_k42.txt, line 1. This row is SmallGroup(70,1) with nonidentity GAP supports a=[2], b=[2, 8, 11]. 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=42), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 2, not an exact certificate. The published distance (d=2) 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(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=[2], b=[2, 8, 11] 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) = 42. 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, 1, 70, 71, 124, 138] [0, 1, 70, 71, 120, 139] [2, 4, 72, 74, 75, 94] [3, 5, 73, 99, 110, 115] [2, 4, 72, 73, 74, 99] [3, 5, 75, 94, 110, 115] [6, 9, 76, 79, 80, 104] [7, 10, 70, 71, 77, 109] [8, 11, 78, 89, 120, 139] [6, 9, 76, 77, 79, 109] [7, 10, 70, 71, 80, 104] [8, 11, 81, 85, 124, 138] [12, 16, 82, 86, 87, 114] [13, 17, 72, 74, 83, 119] [14, 18, 73, 84, 98, 99] [15, 19, 81, 85, 128, 136] [12, 16, 82, 83, 86, 119] [13, 17, 72, 74, 87, 114] [14, 18, 75, 88, 93, 94] [15, 19, 78, 89, 131, 134] [20, 25, 90, 95, 96, 123] [21, 26, 76, 79, 91, 127] [22, 27, 77, 92, 108, 109] [23, 28, 78, 88, 89, 93] [24, 29, 75, 94, 131, 134] [20, 25, 90, 91, 95, 127] [21, 26, 76, 79, 96, 123] [22, 27, 80, 97, 103, 104] [23, 28, 81, 84, 85, 98] [24, 29, 73, 99, 128, 136] [30, 35, 100, 105, 106, 130] [31, 36, 82, 86, 101, 133] [32, 37, 83, 102, 118, 119] [33, 38, 84, 97, 98, 103] [34, 39, 80, 81, 85, 104] [30, 35, 100, 101, 105, 133] [31, 36, 82, 86, 106, 130] [32, 37, 87, 107, 113, 114] [33, 38, 88, 92, 93, 108] [34, 39, 77, 78, 89, 109] [40, 45, 110, 115, 116, 135] [41, 46, 90, 95, 111, 137] [42, 47, 91, 112, 126, 127] [43, 48, 92, 107, 108, 113] [44, 49, 87, 88, 93, 114] [40, 45, 110, 111, 115, 137] [41, 46, 90, 95, 116, 135] [42, 47, 96, 117, 122, 123] [43, 48, 97, 102, 103, 118] [44, 49, 83, 84, 98, 119] [50, 54, 100, 105, 120, 139] [51, 55, 101, 121, 132, 133] [52, 56, 102, 117, 118, 122] [53, 57, 96, 97, 103, 123] [50, 54, 100, 105, 124, 138] [51, 55, 106, 125, 129, 130] [52, 56, 107, 112, 113, 126] [53, 57, 91, 92, 108, 127] [58, 61, 111, 128, 136, 137] [59, 62, 112, 125, 126, 129] [60, 63, 106, 107, 113, 130] [58, 61, 116, 131, 134, 135] [59, 62, 117, 121, 122, 132] [60, 63, 101, 102, 118, 133] [64, 66, 121, 131, 132, 134] [65, 67, 116, 117, 122, 135] [64, 66, 125, 128, 129, 136] [65, 67, 111, 112, 126, 137] [68, 69, 124, 125, 129, 138] [68, 69, 120, 121, 132, 139]
H_Z (70 checks, sparse supports)
[0, 1, 7, 10, 70, 71] [0, 1, 7, 10, 70, 71] [2, 4, 13, 17, 72, 74] [3, 4, 14, 29, 73, 75] [2, 4, 13, 17, 72, 74] [2, 5, 18, 24, 73, 75] [6, 9, 21, 26, 76, 79] [7, 9, 22, 39, 77, 80] [8, 19, 23, 39, 78, 81] [6, 9, 21, 26, 76, 79] [6, 10, 27, 34, 77, 80] [11, 15, 28, 34, 78, 81] [12, 16, 31, 36, 82, 86] [13, 16, 32, 49, 83, 87] [14, 28, 33, 49, 84, 88] [11, 15, 28, 34, 85, 89] [12, 16, 31, 36, 82, 86] [12, 17, 37, 44, 83, 87] [18, 23, 38, 44, 84, 88] [8, 19, 23, 39, 85, 89] [20, 25, 41, 46, 90, 95] [21, 25, 42, 57, 91, 96] [22, 38, 43, 57, 92, 97] [18, 23, 38, 44, 93, 98] [2, 5, 18, 24, 94, 99] [20, 25, 41, 46, 90, 95] [20, 26, 47, 53, 91, 96] [27, 33, 48, 53, 92, 97] [14, 28, 33, 49, 93, 98] [3, 4, 14, 29, 94, 99] [30, 35, 50, 54, 100, 105] [31, 35, 51, 63, 101, 106] [32, 48, 52, 63, 102, 107] [27, 33, 48, 53, 103, 108] [6, 10, 27, 34, 104, 109] [30, 35, 50, 54, 100, 105] [30, 36, 55, 60, 101, 106] [37, 43, 56, 60, 102, 107] [22, 38, 43, 57, 103, 108] [7, 9, 22, 39, 104, 109] [3, 5, 40, 45, 110, 115] [41, 45, 58, 67, 111, 116] [42, 56, 59, 67, 112, 117] [37, 43, 56, 60, 113, 118] [12, 17, 37, 44, 114, 119] [3, 5, 40, 45, 110, 115] [40, 46, 61, 65, 111, 116] [47, 52, 62, 65, 112, 117] [32, 48, 52, 63, 113, 118] [13, 16, 32, 49, 114, 119] [1, 8, 50, 69, 120, 124] [51, 62, 64, 69, 121, 125] [47, 52, 62, 65, 122, 126] [20, 26, 47, 53, 123, 127] [0, 11, 54, 68, 120, 124] [55, 59, 66, 68, 121, 125] [42, 56, 59, 67, 122, 126] [21, 25, 42, 57, 123, 127] [15, 29, 58, 66, 128, 131] [55, 59, 66, 68, 129, 132] [30, 36, 55, 60, 130, 133] [19, 24, 61, 64, 128, 131] [51, 62, 64, 69, 129, 132] [31, 35, 51, 63, 130, 133] [19, 24, 61, 64, 134, 136] [40, 46, 61, 65, 135, 137] [15, 29, 58, 66, 134, 136] [41, 45, 58, 67, 135, 137] [0, 11, 54, 68, 138, 139] [1, 8, 50, 69, 138, 139]
Code ID 140-42-2 · download JSON · raw on GitHub