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[[168,22,7]] d ≤
n
168
k
22
d
7
kd²/n
6.417
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 7, d_Z ≤ 7 · 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 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[8, 19, 31, 43, 55, 67, 77]
d_Z 7 · witness weight 7 (claimed upper_bound)
witness operator (support, 7 qubits)
[93, 104, 116, 128, 140, 152, 162]
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 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)×84 (2,4)×252 (3,3)×84 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 84 (1,4): 84 (2,4): 252 (2,5): 1008 (2,6): 504 (3,3): 84 (3,4): 336 (3,5): 756 (3,6): 8148 (3,7): 13188 (3,8): 3780 (3,9): 1512 (3,10): 336
trapping sets H_Z (1,3)×84 (2,4)×252 (3,3)×84 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 84 (1,4): 84 (2,4): 252 (2,5): 1008 (2,6): 504 (3,3): 84 (3,4): 336 (3,5): 756 (3,6): 8148 (3,7): 13188 (3,8): 3780 (3,9): 1512 (3,10): 336

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(84,2), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt7wtL3_order84_k22.txt, line 3. nonidentity supports a=[3, 20], b=[5, 18, 42].
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 ≤ 8 (computed)

How this code was found

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

[[168,22,7]] — two-block group-algebra code on SmallGroup(84,2)

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 = 6.42.

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_order84_k22.txt, line 3. This row is SmallGroup(84,2) with nonidentity GAP supports a=[3, 20], b=[5, 18, 42]. 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=22), 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(84,2), 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=[3, 20], b=[5, 18, 42] 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) = 22. 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 7) · Z-checks 84 (max weight 7)
H_X (84 checks, sparse supports)
[0, 8, 57, 84, 124, 143, 148] [1, 13, 63, 85, 107, 131, 149] [0, 2, 43, 86, 134, 153, 157] [3, 18, 68, 87, 113, 137, 154] [4, 19, 69, 84, 88, 136, 159] [1, 5, 49, 89, 117, 141, 158] [6, 24, 73, 90, 118, 142, 159] [7, 25, 74, 85, 91, 119, 143] [2, 8, 59, 92, 144, 161, 163] [3, 9, 54, 93, 123, 147, 162] [4, 10, 55, 86, 94, 146, 165] [11, 30, 78, 87, 95, 125, 149] [2, 12, 31, 88, 90, 96, 148] [5, 13, 65, 97, 127, 151, 164] [6, 14, 60, 98, 128, 152, 165] [7, 15, 61, 89, 99, 129, 153] [16, 36, 81, 90, 100, 130, 154] [5, 17, 37, 84, 91, 101, 131] [9, 18, 70, 102, 133, 156, 166] [0, 10, 19, 92, 103, 155, 167] [11, 20, 66, 93, 104, 135, 158] [12, 21, 67, 94, 98, 105, 157] [9, 22, 42, 85, 95, 106, 137] [10, 23, 43, 96, 100, 107, 159] [14, 24, 75, 108, 138, 160, 167] [1, 15, 25, 97, 109, 139, 161] [16, 26, 71, 98, 110, 140, 162] [17, 27, 72, 86, 99, 111, 141] [14, 28, 48, 87, 100, 112, 142] [15, 29, 49, 88, 101, 113, 143] [3, 20, 30, 102, 114, 145, 164] [4, 21, 31, 103, 108, 115, 163] [22, 32, 76, 89, 104, 116, 147] [23, 33, 77, 105, 110, 117, 165] [20, 34, 54, 91, 106, 118, 149] [21, 35, 55, 90, 107, 112, 119] [6, 26, 36, 108, 120, 150, 166] [7, 27, 37, 92, 109, 121, 151] [28, 38, 79, 93, 110, 122, 152] [29, 39, 80, 94, 111, 123, 153] [26, 40, 60, 95, 112, 124, 154] [27, 41, 61, 84, 96, 113, 125] [11, 32, 42, 97, 114, 126, 156] [12, 33, 43, 115, 120, 127, 167] [34, 44, 82, 99, 116, 128, 158] [8, 35, 45, 98, 117, 122, 129] [32, 46, 66, 85, 101, 118, 130] [33, 47, 67, 100, 119, 124, 131] [16, 38, 48, 102, 120, 132, 160] [17, 39, 49, 103, 121, 133, 161] [40, 50, 83, 104, 122, 134, 162] [13, 41, 51, 86, 105, 123, 135] [38, 52, 71, 87, 106, 124, 136] [39, 53, 72, 88, 107, 125, 137] [22, 44, 54, 109, 126, 138, 164] [23, 45, 55, 108, 127, 132, 139] [18, 46, 56, 89, 111, 128, 140] [19, 47, 57, 110, 129, 134, 141] [44, 58, 76, 91, 113, 130, 142] [45, 59, 77, 112, 131, 136, 143] [28, 50, 60, 114, 132, 144, 166] [29, 51, 61, 92, 115, 133, 145] [24, 52, 62, 93, 116, 134, 146] [25, 53, 63, 94, 117, 135, 147] [50, 64, 79, 95, 118, 136, 148] [51, 65, 80, 96, 119, 137, 149] [34, 56, 66, 97, 121, 138, 150] [35, 57, 67, 120, 139, 144, 151] [30, 58, 68, 99, 123, 140, 152] [31, 59, 69, 122, 141, 146, 153] [56, 70, 82, 101, 125, 142, 154] [40, 62, 71, 102, 126, 144, 155] [41, 63, 72, 103, 127, 145, 156] [36, 64, 73, 104, 128, 146, 157] [37, 65, 74, 105, 129, 147, 158] [62, 75, 83, 106, 130, 148, 159] [46, 68, 76, 109, 133, 150, 160] [47, 69, 77, 132, 151, 155, 161] [42, 70, 78, 111, 135, 152, 162] [52, 73, 79, 114, 138, 155, 163] [53, 74, 80, 115, 139, 156, 164] [48, 75, 81, 116, 140, 157, 165] [58, 78, 82, 121, 145, 160, 166] [64, 81, 83, 126, 150, 163, 167]
H_Z (84 checks, sparse supports)
[0, 4, 17, 41, 84, 86, 103] [1, 7, 22, 46, 85, 89, 109] [2, 10, 27, 51, 86, 92, 96] [3, 11, 28, 52, 87, 93, 114] [4, 12, 29, 53, 88, 94, 115] [5, 15, 32, 56, 89, 97, 101] [6, 12, 16, 35, 90, 98, 120] [7, 17, 34, 58, 91, 99, 121] [8, 19, 37, 61, 84, 92, 129] [9, 20, 38, 62, 93, 102, 106] [10, 21, 39, 63, 94, 103, 107] [11, 22, 40, 64, 95, 104, 126] [12, 23, 41, 65, 96, 105, 127] [13, 25, 42, 66, 85, 97, 135] [14, 21, 26, 45, 98, 108, 112] [15, 27, 44, 68, 99, 109, 113] [16, 23, 28, 47, 100, 110, 132] [17, 29, 46, 70, 101, 111, 133] [18, 30, 48, 71, 87, 102, 140] [19, 31, 49, 72, 88, 103, 141] [20, 32, 50, 73, 104, 114, 118] [21, 33, 51, 74, 105, 115, 119] [22, 34, 52, 75, 106, 116, 138] [1, 23, 35, 53, 107, 117, 139] [24, 31, 36, 55, 90, 108, 146] [25, 37, 54, 76, 91, 109, 147] [26, 33, 38, 57, 110, 120, 124] [27, 39, 56, 78, 111, 121, 125] [28, 35, 40, 59, 112, 122, 144] [3, 29, 41, 58, 113, 123, 145] [30, 42, 60, 79, 95, 114, 152] [31, 43, 61, 80, 96, 115, 153] [32, 44, 62, 81, 116, 126, 130] [5, 33, 45, 63, 117, 127, 131] [6, 34, 46, 64, 118, 128, 150] [7, 35, 47, 65, 119, 129, 151] [36, 43, 48, 67, 100, 120, 157] [37, 49, 66, 82, 101, 121, 158] [38, 45, 50, 69, 122, 132, 136] [9, 39, 51, 68, 123, 133, 137] [0, 40, 47, 52, 124, 134, 155] [11, 41, 53, 70, 125, 135, 156] [42, 54, 71, 83, 106, 126, 162] [13, 43, 55, 72, 86, 107, 127] [14, 44, 56, 73, 128, 138, 142] [15, 45, 57, 74, 129, 139, 143] [16, 46, 58, 75, 130, 140, 160] [1, 17, 47, 59, 131, 141, 161] [48, 55, 60, 77, 112, 132, 165] [18, 49, 61, 76, 89, 113, 133] [2, 50, 57, 62, 134, 144, 148] [20, 51, 63, 78, 135, 145, 149] [4, 52, 59, 64, 136, 146, 163] [3, 22, 53, 65, 137, 147, 164] [24, 54, 66, 79, 93, 118, 138] [25, 55, 67, 80, 94, 119, 139] [26, 56, 68, 81, 140, 150, 154] [5, 27, 57, 69, 84, 141, 151] [6, 28, 58, 70, 142, 152, 166] [0, 7, 29, 59, 92, 143, 153] [8, 60, 67, 71, 98, 124, 144] [30, 61, 72, 82, 99, 125, 145] [10, 62, 69, 73, 146, 155, 159] [9, 32, 63, 74, 85, 147, 156] [0, 12, 64, 75, 148, 157, 167] [1, 11, 34, 65, 97, 149, 158] [36, 66, 76, 83, 104, 130, 150] [13, 37, 67, 77, 105, 131, 151] [14, 38, 68, 78, 87, 152, 160] [2, 15, 39, 69, 88, 153, 161] [3, 16, 40, 70, 102, 154, 162] [19, 71, 77, 79, 110, 136, 155] [18, 42, 72, 80, 111, 137, 156] [2, 21, 73, 81, 90, 157, 163] [5, 20, 44, 74, 91, 158, 164] [4, 6, 23, 75, 108, 159, 165] [24, 48, 76, 82, 116, 142, 160] [8, 25, 49, 77, 117, 143, 161] [9, 26, 50, 78, 95, 162, 166] [8, 31, 79, 83, 122, 148, 163] [13, 30, 54, 80, 123, 149, 164] [10, 14, 33, 81, 100, 165, 167] [18, 36, 60, 82, 128, 154, 166] [19, 24, 43, 83, 134, 159, 167]
Code ID 168-22-7 · download JSON · raw on GitHub