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[[168,16,10]] d ≤
n
168
k
16
d
10
kd²/n
9.524
w
6
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 10, d_Z ≤ 10 · 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 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[15, 25, 29, 48, 74, 95, 137, 138, 152, 164]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[23, 42, 61, 67, 74, 85, 100, 122, 137, 144]
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 3 · H_Z 3
trapping sets H_X (1,3)×168 (2,4)×1260 (3,3)×126 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 168 (2,4): 1260 (3,3): 126 (3,5): 12222 (3,7): 1680
trapping sets H_Z (1,3)×168 (2,4)×1260 (3,3)×126 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 168 (2,4): 1260 (3,3): 126 (3,5): 12222 (3,7): 1680

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

[[168,16,10]] — two-block group-algebra code on SmallGroup(84,13)

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 = 9.52.

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_wt6wtL3_order84_k16.txt, line 13. This row is SmallGroup(84,13) with nonidentity GAP supports a=[7, 44], b=[12, 48]. 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=16), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 10, not an exact certificate. The published distance (d=10) 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(84,13), 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=[7, 44] and b=[12, 48] 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) = 16. 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 6) · Z-checks 84 (max weight 6)
H_X (84 checks, sparse supports)
[0, 63, 68, 84, 130, 162] [1, 57, 61, 85, 136, 165] [2, 59, 71, 86, 139, 166] [1, 3, 77, 87, 96, 142] [4, 42, 74, 88, 141, 143] [5, 52, 66, 89, 145, 167] [0, 6, 72, 90, 101, 148] [7, 69, 73, 91, 147, 149] [5, 8, 70, 92, 104, 151] [9, 33, 79, 93, 150, 152] [6, 10, 82, 94, 107, 153] [7, 11, 54, 84, 95, 154] [12, 55, 81, 96, 117, 153] [2, 13, 64, 97, 110, 156] [14, 65, 76, 98, 155, 157] [3, 15, 79, 99, 113, 158] [4, 16, 80, 85, 100, 159] [17, 48, 78, 101, 123, 158] [13, 18, 78, 102, 116, 160] [14, 19, 45, 86, 103, 161] [20, 46, 83, 104, 126, 160] [15, 20, 21, 88, 105, 119] [16, 22, 66, 87, 106, 162] [17, 23, 67, 88, 107, 129] [8, 24, 74, 108, 122, 163] [9, 25, 75, 89, 109, 164] [26, 39, 82, 110, 132, 163] [10, 14, 27, 91, 111, 125] [11, 28, 83, 90, 112, 165] [12, 29, 60, 91, 113, 135] [12, 24, 30, 93, 114, 128] [25, 31, 57, 92, 115, 166] [26, 32, 58, 93, 116, 138] [27, 32, 33, 95, 117, 131] [2, 28, 34, 94, 96, 118] [29, 35, 76, 95, 119, 141] [7, 18, 36, 98, 120, 134] [19, 37, 81, 97, 121, 167] [20, 38, 51, 98, 122, 144] [21, 25, 39, 100, 123, 137] [22, 26, 40, 99, 101, 124] [23, 41, 71, 100, 125, 147] [23, 36, 42, 103, 126, 140] [0, 37, 43, 102, 104, 127] [38, 44, 69, 103, 128, 150] [39, 44, 45, 106, 129, 143] [8, 40, 46, 105, 107, 130] [9, 41, 47, 84, 106, 131] [16, 30, 48, 109, 132, 146] [17, 31, 49, 108, 110, 133] [32, 50, 63, 109, 134, 155] [33, 37, 51, 112, 135, 149] [34, 38, 52, 111, 113, 136] [5, 35, 53, 85, 112, 137] [35, 48, 54, 115, 138, 152] [3, 49, 55, 114, 116, 139] [4, 50, 56, 86, 115, 140] [51, 56, 57, 118, 141, 154] [18, 52, 58, 117, 119, 142] [19, 53, 59, 87, 118, 143] [28, 42, 60, 121, 144, 157] [29, 43, 61, 120, 122, 145] [1, 44, 62, 89, 121, 146] [45, 49, 63, 124, 147, 159] [46, 50, 64, 123, 125, 148] [13, 47, 65, 90, 124, 149] [47, 60, 66, 127, 150, 161] [10, 61, 67, 126, 128, 151] [11, 62, 68, 92, 127, 152] [30, 64, 69, 129, 131, 153] [31, 65, 70, 94, 130, 154] [40, 54, 71, 133, 155, 164] [41, 55, 72, 132, 134, 156] [6, 56, 73, 97, 133, 157] [58, 62, 74, 135, 137, 158] [24, 59, 75, 99, 136, 159] [21, 72, 76, 138, 140, 160] [22, 73, 77, 102, 139, 161] [43, 75, 78, 105, 142, 162] [53, 67, 79, 144, 146, 163] [15, 68, 80, 108, 145, 164] [36, 70, 81, 111, 148, 165] [34, 80, 82, 114, 151, 166] [27, 77, 83, 120, 156, 167]
H_Z (84 checks, sparse supports)
[0, 11, 47, 84, 90, 127] [1, 16, 53, 85, 87, 146] [2, 19, 56, 86, 97, 118] [3, 22, 59, 87, 99, 139] [4, 21, 23, 88, 100, 140] [5, 25, 62, 89, 92, 137] [6, 28, 65, 90, 94, 157] [7, 27, 29, 91, 95, 120] [8, 31, 68, 92, 108, 130] [9, 30, 32, 93, 109, 131] [10, 34, 70, 94, 111, 151] [11, 33, 35, 95, 112, 152] [3, 12, 34, 96, 113, 114] [13, 37, 73, 97, 102, 149] [14, 36, 38, 98, 103, 111] [15, 40, 75, 99, 105, 164] [16, 39, 41, 100, 106, 132] [6, 17, 40, 101, 107, 133] [18, 43, 77, 102, 120, 142] [19, 42, 44, 103, 121, 143] [8, 20, 43, 104, 105, 122] [21, 46, 78, 105, 123, 160] [22, 45, 47, 106, 124, 161] [10, 23, 46, 107, 125, 126] [24, 49, 80, 108, 114, 159] [25, 48, 50, 109, 115, 123] [13, 26, 49, 110, 116, 124] [27, 52, 81, 111, 117, 167] [28, 51, 53, 112, 118, 144] [15, 29, 52, 113, 119, 145] [30, 55, 82, 114, 132, 153] [31, 54, 56, 115, 133, 154] [18, 32, 55, 116, 117, 134] [12, 33, 58, 93, 117, 135] [34, 57, 59, 118, 136, 166] [21, 35, 58, 119, 137, 138] [36, 61, 83, 120, 126, 165] [37, 60, 62, 121, 127, 135] [24, 38, 61, 122, 128, 136] [17, 39, 64, 110, 123, 129] [40, 63, 65, 124, 130, 155] [27, 41, 64, 125, 131, 156] [20, 42, 67, 88, 126, 144] [43, 66, 68, 127, 145, 162] [30, 44, 67, 128, 129, 146] [23, 45, 69, 103, 129, 147] [0, 46, 70, 104, 130, 148] [33, 47, 69, 131, 149, 150] [26, 48, 72, 101, 132, 138] [49, 71, 73, 133, 139, 147] [36, 50, 72, 134, 140, 148] [29, 51, 74, 122, 135, 141] [1, 52, 75, 89, 136, 142] [39, 53, 74, 137, 143, 163] [32, 54, 76, 95, 138, 155] [2, 55, 77, 96, 139, 156] [42, 56, 76, 140, 141, 157] [4, 35, 57, 85, 115, 141] [3, 58, 78, 116, 142, 158] [4, 45, 59, 86, 143, 159] [38, 60, 79, 113, 144, 150] [5, 61, 80, 85, 145, 151] [48, 62, 79, 146, 152, 158] [7, 41, 63, 84, 134, 147] [6, 64, 81, 97, 148, 153] [7, 51, 65, 98, 149, 154] [9, 44, 66, 89, 106, 150] [8, 67, 82, 107, 151, 163] [9, 54, 68, 84, 152, 164] [10, 12, 69, 91, 128, 153] [11, 57, 70, 92, 154, 165] [14, 50, 71, 86, 125, 155] [13, 72, 83, 90, 156, 160] [14, 60, 73, 91, 157, 161] [15, 17, 74, 88, 108, 158] [16, 63, 75, 109, 159, 162] [18, 20, 76, 98, 119, 160] [19, 66, 77, 87, 161, 167] [0, 22, 78, 101, 102, 162] [24, 26, 79, 93, 99, 163] [25, 71, 80, 100, 164, 166] [1, 28, 81, 96, 121, 165] [2, 31, 82, 94, 110, 166] [5, 37, 83, 104, 112, 167]
Code ID 168-16-10 · download JSON · raw on GitHub