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[[168,8,7]] d ≤
n
168
k
8
d
7
kd²/n
2.333
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 7, d_Z ≤ 7 · w_X = 5, w_Z = 5 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[84, 86, 92, 102, 114, 126, 138]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[11, 21, 33, 45, 57, 68, 76]
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 8 · H_Z 8 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 5 · H_Z 5
qubit degrees H_X 2–3 (mean 2.5) · H_Z 2–3 (mean 2.5)
trapping sets H_X (1,2)×84 (2,2)×84 (3,2)×84 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 84 (1,3): 84 (2,2): 84 (2,3): 504 (2,4): 252 (3,2): 84 (3,3): 1512 (3,4): 2772 (3,5): 1260 (3,6): 504 (3,7): 84
trapping sets H_Z (1,2)×84 (2,2)×84 (3,2)×84 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 84 (1,3): 84 (2,2): 84 (2,3): 504 (2,4): 252 (3,2): 84 (3,3): 1512 (3,4): 2772 (3,5): 1260 (3,6): 504 (3,7): 84

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(84,3), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt5wtL2_order84_k8.txt, line 5. nonidentity supports a=[3], b=[2, 22].
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

[[168,8,7]] — two-block group-algebra code on SmallGroup(84,3)

Direction & hypothesis

Target: unrestricted weight-6 frontier for hackathon issue #1155. Reconstruct a published weight-5 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 2.33.

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_wt5wtL2_order84_k8.txt, line 5. This row is SmallGroup(84,3) with nonidentity GAP supports a=[3], b=[2, 22]. 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 ≤ 7, not an exact certificate. The published distance (d=7) 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(84,3), 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], b=[2, 22] 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 = 168 - 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 84 (max weight 5) · Z-checks 84 (max weight 5)
H_X (84 checks, sparse supports)
[0, 54, 84, 90, 166] [1, 60, 84, 85, 167] [0, 2, 86, 98, 107] [3, 66, 85, 87, 161] [4, 67, 88, 113, 150] [1, 5, 86, 89, 113] [6, 71, 87, 90, 164] [7, 72, 91, 96, 155] [2, 8, 92, 109, 119] [3, 9, 89, 93, 96] [4, 10, 87, 94, 125] [11, 76, 95, 101, 138] [12, 77, 96, 100, 160] [5, 13, 92, 97, 125] [6, 14, 93, 98, 101] [7, 15, 90, 99, 106] [16, 79, 100, 107, 144] [17, 80, 88, 101, 163] [8, 18, 102, 121, 131] [9, 19, 97, 103, 106] [10, 20, 93, 104, 137] [11, 21, 84, 105, 112] [12, 22, 95, 106, 111] [23, 82, 91, 107, 151] [13, 24, 102, 108, 137] [14, 25, 103, 109, 112] [15, 26, 98, 110, 118] [16, 27, 85, 111, 119] [17, 28, 94, 100, 112] [29, 83, 95, 113, 156] [18, 30, 114, 133, 143] [19, 31, 108, 115, 118] [20, 32, 103, 116, 149] [21, 33, 86, 117, 124] [22, 34, 105, 118, 123] [23, 35, 88, 99, 119] [24, 36, 114, 120, 149] [25, 37, 115, 121, 124] [26, 38, 109, 122, 130] [27, 39, 89, 123, 131] [28, 40, 104, 111, 124] [29, 41, 91, 105, 125] [30, 42, 126, 145, 154] [31, 43, 120, 127, 130] [32, 44, 115, 128, 159] [33, 45, 92, 129, 136] [34, 46, 117, 130, 135] [35, 47, 94, 110, 131] [36, 48, 126, 132, 159] [37, 49, 127, 133, 136] [38, 50, 121, 134, 142] [39, 51, 97, 135, 143] [40, 52, 116, 123, 136] [41, 53, 99, 117, 137] [42, 54, 138, 155, 162] [43, 55, 132, 139, 142] [44, 56, 127, 140, 165] [45, 57, 102, 141, 148] [46, 58, 129, 142, 147] [47, 59, 104, 122, 143] [48, 60, 138, 144, 165] [49, 61, 139, 145, 148] [50, 62, 133, 146, 153] [51, 63, 108, 147, 154] [52, 64, 128, 135, 148] [53, 65, 110, 129, 149] [55, 66, 144, 150, 153] [56, 67, 139, 151, 167] [57, 68, 114, 152, 158] [58, 69, 141, 153, 157] [59, 70, 116, 134, 154] [61, 71, 150, 155, 158] [62, 72, 145, 156, 161] [63, 73, 120, 157, 162] [64, 74, 140, 147, 158] [65, 75, 122, 141, 159] [68, 76, 126, 160, 164] [69, 77, 152, 161, 163] [70, 78, 128, 146, 162] [73, 79, 132, 163, 166] [74, 80, 151, 157, 164] [75, 81, 134, 152, 165] [78, 82, 140, 156, 166] [81, 83, 146, 160, 167]
H_Z (84 checks, sparse supports)
[0, 1, 21, 84, 86] [1, 3, 27, 85, 89] [2, 5, 33, 86, 92] [3, 6, 10, 87, 93] [4, 17, 35, 88, 94] [5, 9, 39, 89, 97] [0, 6, 15, 90, 98] [7, 23, 41, 91, 99] [8, 13, 45, 92, 102] [9, 14, 20, 93, 103] [10, 28, 47, 94, 104] [11, 22, 29, 95, 105] [7, 9, 12, 96, 106] [13, 19, 51, 97, 108] [2, 14, 26, 98, 109] [15, 35, 53, 99, 110] [12, 16, 28, 100, 111] [11, 14, 17, 101, 112] [18, 24, 57, 102, 114] [19, 25, 32, 103, 115] [20, 40, 59, 104, 116] [21, 34, 41, 105, 117] [15, 19, 22, 106, 118] [2, 16, 23, 107, 119] [24, 31, 63, 108, 120] [8, 25, 38, 109, 121] [26, 47, 65, 110, 122] [22, 27, 40, 111, 123] [21, 25, 28, 112, 124] [4, 5, 29, 113, 125] [30, 36, 68, 114, 126] [31, 37, 44, 115, 127] [32, 52, 70, 116, 128] [33, 46, 53, 117, 129] [26, 31, 34, 118, 130] [8, 27, 35, 119, 131] [36, 43, 73, 120, 132] [18, 37, 50, 121, 133] [38, 59, 75, 122, 134] [34, 39, 52, 123, 135] [33, 37, 40, 124, 136] [10, 13, 41, 125, 137] [42, 48, 76, 126, 138] [43, 49, 56, 127, 139] [44, 64, 78, 128, 140] [45, 58, 65, 129, 141] [38, 43, 46, 130, 142] [18, 39, 47, 131, 143] [48, 55, 79, 132, 144] [30, 49, 62, 133, 145] [50, 70, 81, 134, 146] [46, 51, 64, 135, 147] [45, 49, 52, 136, 148] [20, 24, 53, 137, 149] [11, 54, 60, 84, 138] [55, 61, 67, 139, 150] [56, 74, 82, 140, 151] [57, 69, 75, 141, 152] [50, 55, 58, 142, 153] [30, 51, 59, 143, 154] [16, 60, 66, 85, 144] [42, 61, 72, 145, 155] [62, 78, 83, 146, 156] [58, 63, 74, 147, 157] [57, 61, 64, 148, 158] [32, 36, 65, 149, 159] [4, 66, 71, 87, 150] [23, 67, 80, 88, 151] [68, 77, 81, 152, 160] [62, 66, 69, 153, 161] [42, 63, 70, 154, 162] [7, 54, 71, 90, 155] [29, 72, 82, 91, 156] [69, 73, 80, 157, 163] [68, 71, 74, 158, 164] [44, 48, 75, 159, 165] [12, 76, 83, 95, 160] [3, 72, 77, 96, 161] [54, 73, 78, 162, 166] [17, 77, 79, 100, 163] [6, 76, 80, 101, 164] [56, 60, 81, 165, 167] [0, 79, 82, 107, 166] [1, 67, 83, 113, 167]
Code ID 168-8-7 · download JSON · raw on GitHub