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[[168,4,14]] d ≤
n
168
k
4
d
14
kd²/n
4.667
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 14, d_Z ≤ 14 · 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 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[9, 10, 34, 56, 58, 66, 89, 103, 114, 118, 121, 125, 142, 149]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[8, 17, 30, 35, 53, 80, 85, 99, 123, 124, 130, 133, 160, 164]
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 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): 1596 (3,4): 2772 (3,5): 1008 (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): 1596 (3,4): 2772 (3,5): 1008 (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_k4.txt, line 14. nonidentity supports a=[6], b=[11, 60].
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

[[168,4,14]] — 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 = 4.67.

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_k4.txt, line 14. This row is SmallGroup(84,3) with nonidentity GAP supports a=[6], b=[11, 60]. 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=4), 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

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=[6], b=[11, 60] 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) = 4. 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, 71, 84, 141, 161] [1, 54, 85, 147, 164] [2, 6, 86, 96, 152] [3, 60, 87, 128, 166] [4, 79, 88, 138, 154] [0, 5, 89, 101, 157] [6, 66, 90, 134, 167] [7, 67, 91, 144, 159] [8, 14, 92, 106, 160] [1, 9, 93, 107, 140] [10, 16, 84, 94, 162] [11, 72, 95, 142, 150] [12, 83, 96, 127, 151] [2, 13, 97, 112, 163] [3, 14, 98, 113, 146] [4, 15, 85, 99, 165] [16, 76, 100, 148, 155] [17, 77, 101, 133, 156] [18, 25, 95, 102, 118] [5, 19, 103, 119, 151] [20, 27, 86, 104, 166] [7, 21, 87, 105, 153] [22, 29, 88, 106, 139] [23, 80, 107, 114, 160] [8, 24, 100, 108, 124] [9, 25, 109, 125, 156] [10, 26, 89, 110, 167] [11, 27, 90, 111, 158] [12, 28, 91, 112, 145] [29, 82, 113, 120, 163] [30, 37, 105, 114, 130] [13, 31, 88, 115, 131] [32, 39, 92, 107, 116] [15, 33, 93, 117, 161] [34, 41, 94, 118, 150] [17, 35, 95, 119, 126] [18, 36, 111, 120, 136] [19, 37, 91, 121, 137] [20, 38, 97, 113, 122] [21, 39, 98, 123, 164] [22, 40, 99, 124, 155] [23, 41, 100, 125, 132] [42, 49, 117, 126, 142] [24, 43, 94, 127, 143] [44, 51, 102, 119, 128] [26, 45, 96, 103, 129] [46, 53, 87, 104, 130] [28, 47, 105, 131, 138] [30, 48, 123, 132, 148] [31, 49, 99, 133, 149] [32, 50, 108, 125, 134] [33, 51, 101, 109, 135] [34, 52, 90, 110, 136] [35, 53, 111, 137, 144] [54, 61, 129, 138, 153] [36, 55, 104, 139, 154] [56, 63, 114, 131, 140] [38, 57, 106, 115, 141] [58, 65, 93, 116, 142] [40, 59, 84, 117, 143] [42, 60, 135, 144, 158] [43, 61, 110, 145, 159] [44, 62, 120, 137, 146] [45, 63, 112, 121, 147] [46, 64, 98, 122, 148] [47, 65, 85, 123, 149] [48, 66, 116, 150, 162] [67, 73, 126, 143, 151] [50, 68, 118, 127, 152] [69, 75, 103, 128, 153] [52, 70, 86, 129, 154] [55, 71, 122, 155, 165] [56, 72, 132, 149, 156] [57, 73, 124, 133, 157] [58, 74, 109, 134, 158] [59, 75, 89, 135, 159] [62, 76, 130, 139, 160] [77, 81, 115, 140, 161] [64, 78, 92, 141, 162] [68, 79, 136, 145, 163] [69, 80, 121, 146, 164] [70, 81, 97, 147, 165] [74, 82, 102, 152, 166] [78, 83, 108, 157, 167]
H_Z (84 checks, sparse supports)
[0, 10, 59, 84, 89] [1, 15, 65, 85, 93] [2, 20, 70, 86, 97] [3, 21, 46, 87, 98] [4, 22, 31, 88, 99] [5, 26, 75, 89, 103] [6, 27, 52, 86, 90] [7, 28, 37, 91, 105] [8, 32, 78, 92, 108] [9, 33, 58, 93, 109] [10, 34, 43, 94, 110] [11, 18, 35, 95, 111] [2, 12, 45, 96, 112] [13, 38, 81, 97, 115] [14, 39, 64, 92, 98] [15, 40, 49, 99, 117] [16, 24, 41, 94, 100] [5, 17, 51, 101, 119] [18, 44, 82, 102, 120] [19, 45, 69, 103, 121] [20, 46, 55, 104, 122] [21, 30, 47, 105, 123] [8, 22, 57, 106, 124] [9, 23, 32, 107, 125] [24, 50, 83, 108, 127] [25, 51, 74, 102, 109] [26, 52, 61, 110, 129] [27, 36, 53, 104, 111] [13, 28, 63, 112, 131] [14, 29, 38, 106, 113] [23, 30, 56, 114, 132] [31, 57, 77, 115, 133] [32, 58, 66, 116, 134] [33, 42, 59, 117, 135] [18, 34, 68, 118, 136] [19, 35, 44, 119, 137] [29, 36, 62, 120, 139] [37, 63, 80, 114, 121] [38, 64, 71, 122, 141] [39, 48, 65, 116, 123] [24, 40, 73, 124, 143] [25, 41, 50, 118, 125] [35, 42, 67, 126, 144] [12, 43, 68, 127, 145] [3, 44, 69, 128, 146] [45, 54, 70, 129, 147] [30, 46, 76, 130, 148] [31, 47, 56, 131, 149] [41, 48, 72, 132, 150] [17, 49, 73, 126, 133] [6, 50, 74, 134, 152] [51, 60, 75, 128, 135] [36, 52, 79, 136, 154] [37, 53, 62, 130, 137] [4, 47, 54, 85, 138] [22, 55, 76, 139, 155] [9, 56, 77, 140, 156] [0, 57, 78, 141, 157] [11, 42, 58, 142, 158] [43, 59, 67, 143, 159] [7, 53, 60, 87, 144] [28, 61, 79, 138, 145] [14, 62, 80, 146, 160] [1, 63, 81, 140, 147] [16, 48, 64, 148, 162] [49, 65, 72, 142, 149] [11, 34, 66, 90, 150] [12, 19, 67, 91, 151] [2, 68, 82, 152, 163] [21, 54, 69, 153, 164] [4, 55, 70, 154, 165] [16, 40, 71, 84, 155] [17, 25, 72, 95, 156] [5, 73, 83, 151, 157] [27, 60, 74, 158, 166] [7, 61, 75, 153, 159] [8, 23, 76, 100, 160] [0, 33, 77, 101, 161] [10, 66, 78, 162, 167] [13, 29, 79, 88, 163] [1, 39, 80, 107, 164] [15, 71, 81, 161, 165] [3, 20, 82, 113, 166] [6, 26, 83, 96, 167]
Code ID 168-4-14-b · download JSON · raw on GitHub