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[[168,14,10]] d ≤
n
168
k
14
d
10
kd²/n
8.333
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)
[3, 11, 19, 33, 35, 59, 66, 70, 76, 78]
d_Z 10 · witness weight 10 (claimed upper_bound)
witness operator (support, 10 qubits)
[85, 89, 91, 99, 124, 132, 136, 146, 148, 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 6 · H_Z 6
qubit degrees H_X 3 · H_Z 3
trapping sets H_X (1,3)×168 (2,4)×1260 (3,3)×210 (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): 210 (3,5): 11970 (3,7): 1680
trapping sets H_Z (1,3)×168 (2,4)×1260 (3,3)×210 (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): 210 (3,5): 11970 (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

[[168,14,10]] — two-block group-algebra code on SmallGroup(84,3)

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 = 8.33.

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_wt6wtL3_order84_k14.txt, line 1. This row is SmallGroup(84,3) with nonidentity GAP supports a=[6, 46], b=[11, 47]. 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=14), 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(84,3), 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, 46] and b=[11, 47] 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) = 14. 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, 70, 71, 84, 128, 161] [1, 54, 63, 85, 134, 164] [2, 6, 78, 86, 96, 140] [3, 58, 60, 87, 141, 166] [4, 43, 79, 88, 138, 142] [0, 5, 73, 89, 101, 146] [6, 50, 66, 90, 147, 167] [7, 67, 75, 91, 144, 148] [8, 14, 82, 92, 106, 151] [1, 9, 69, 93, 107, 152] [10, 16, 55, 84, 94, 153] [11, 30, 72, 95, 150, 154] [12, 57, 83, 96, 114, 151] [2, 13, 79, 97, 112, 156] [3, 14, 62, 98, 113, 157] [4, 15, 81, 85, 99, 158] [16, 64, 76, 100, 155, 159] [17, 49, 77, 101, 120, 156] [18, 23, 25, 88, 102, 118] [5, 19, 77, 103, 119, 160] [20, 27, 66, 86, 104, 161] [7, 21, 42, 87, 105, 162] [22, 29, 68, 88, 106, 126] [23, 44, 80, 107, 127, 160] [8, 16, 24, 91, 108, 124] [9, 25, 72, 109, 125, 163] [10, 26, 83, 89, 110, 164] [11, 27, 74, 90, 111, 165] [12, 28, 61, 91, 112, 132] [29, 36, 82, 113, 133, 163] [30, 35, 37, 94, 114, 130] [12, 13, 31, 95, 115, 131] [3, 32, 39, 92, 96, 116] [15, 33, 54, 93, 117, 166] [34, 41, 76, 94, 118, 138] [17, 35, 56, 95, 119, 139] [18, 27, 36, 99, 120, 136] [7, 19, 37, 100, 121, 137] [20, 29, 38, 97, 101, 122] [21, 39, 80, 98, 123, 167] [22, 40, 71, 99, 124, 144] [23, 41, 48, 100, 125, 145] [42, 47, 49, 104, 126, 142] [22, 24, 43, 105, 127, 143] [9, 44, 51, 102, 106, 128] [0, 26, 45, 103, 107, 129] [11, 46, 53, 84, 104, 130] [28, 47, 67, 105, 131, 150] [30, 39, 48, 110, 132, 148] [15, 31, 49, 111, 133, 149] [32, 41, 50, 108, 112, 134] [17, 33, 51, 109, 113, 135] [6, 34, 52, 85, 110, 136] [35, 53, 60, 111, 137, 155] [54, 59, 61, 116, 138, 153] [34, 36, 55, 117, 139, 154] [19, 56, 63, 114, 118, 140] [2, 38, 57, 115, 119, 141] [21, 58, 65, 86, 116, 142] [4, 40, 59, 87, 117, 143] [42, 51, 60, 122, 144, 158] [26, 43, 61, 123, 145, 159] [44, 53, 62, 120, 124, 146] [28, 45, 63, 121, 125, 147] [14, 46, 64, 89, 122, 148] [1, 47, 65, 90, 123, 149] [46, 48, 66, 129, 150, 162] [31, 67, 73, 126, 130, 151] [8, 50, 68, 127, 131, 152] [33, 69, 75, 92, 128, 153] [10, 52, 70, 93, 129, 154] [38, 55, 71, 135, 155, 165] [56, 65, 72, 132, 136, 156] [40, 57, 73, 133, 137, 157] [25, 58, 74, 97, 134, 158] [5, 59, 75, 98, 135, 159] [18, 62, 76, 139, 143, 160] [45, 77, 81, 102, 140, 161] [20, 64, 78, 103, 141, 162] [52, 68, 79, 145, 149, 163] [37, 69, 80, 108, 146, 164] [13, 70, 81, 109, 147, 165] [32, 74, 82, 115, 152, 166] [24, 78, 83, 121, 157, 167]
H_Z (84 checks, sparse supports)
[0, 10, 46, 84, 89, 129] [1, 15, 52, 85, 93, 149] [2, 20, 58, 86, 97, 141] [3, 21, 59, 87, 98, 116] [4, 18, 22, 88, 99, 143] [5, 26, 64, 89, 103, 159] [6, 27, 65, 86, 90, 136] [7, 24, 28, 91, 105, 121] [8, 32, 69, 92, 108, 152] [9, 33, 70, 93, 109, 128] [10, 30, 34, 94, 110, 154] [11, 31, 35, 95, 111, 130] [2, 12, 32, 96, 112, 115] [13, 38, 74, 97, 115, 165] [14, 39, 75, 92, 98, 148] [15, 36, 40, 99, 117, 133] [16, 37, 41, 94, 100, 108] [5, 17, 38, 101, 119, 135] [18, 44, 77, 102, 120, 160] [19, 45, 78, 103, 121, 140] [20, 42, 46, 104, 122, 162] [21, 43, 47, 105, 123, 142] [8, 22, 44, 106, 124, 127] [9, 23, 45, 102, 107, 125] [24, 50, 80, 108, 127, 167] [25, 51, 81, 102, 109, 158] [26, 48, 52, 110, 129, 145] [27, 49, 53, 104, 111, 120] [13, 28, 50, 112, 131, 147] [14, 29, 51, 106, 113, 122] [12, 30, 56, 95, 114, 132] [31, 57, 82, 115, 133, 151] [32, 54, 58, 116, 134, 166] [33, 55, 59, 117, 135, 153] [18, 34, 56, 118, 136, 139] [19, 35, 57, 114, 119, 137] [17, 36, 62, 113, 120, 139] [37, 63, 83, 114, 121, 164] [38, 60, 64, 122, 141, 155] [39, 61, 65, 116, 123, 132] [24, 40, 62, 124, 143, 157] [25, 41, 63, 118, 125, 134] [22, 42, 67, 105, 126, 144] [23, 43, 68, 88, 127, 145] [0, 44, 69, 107, 128, 146] [45, 66, 70, 129, 147, 161] [30, 46, 67, 130, 148, 150] [31, 47, 68, 126, 131, 149] [28, 48, 72, 125, 132, 150] [29, 49, 73, 101, 126, 133] [1, 50, 74, 90, 134, 152] [51, 71, 75, 128, 135, 144] [36, 52, 72, 136, 154, 163] [37, 53, 73, 130, 137, 146] [4, 34, 54, 85, 117, 138] [35, 55, 76, 94, 139, 155] [2, 56, 77, 119, 140, 156] [3, 57, 78, 96, 141, 157] [4, 42, 58, 87, 142, 158] [43, 59, 76, 138, 143, 159] [7, 40, 60, 87, 137, 144] [41, 61, 79, 112, 138, 145] [5, 62, 80, 98, 146, 160] [6, 63, 81, 85, 140, 147] [7, 48, 64, 100, 148, 162] [49, 65, 79, 142, 149, 156] [11, 47, 66, 90, 104, 150] [8, 12, 67, 91, 131, 151] [9, 68, 82, 106, 152, 163] [10, 54, 69, 93, 153, 164] [11, 55, 70, 84, 154, 165] [16, 53, 71, 84, 124, 155] [13, 17, 72, 95, 109, 156] [14, 73, 83, 89, 151, 157] [15, 60, 74, 111, 158, 166] [16, 61, 75, 91, 153, 159] [19, 23, 76, 100, 118, 160] [0, 20, 77, 101, 103, 161] [21, 66, 78, 86, 162, 167] [25, 29, 79, 88, 97, 163] [1, 26, 80, 107, 123, 164] [27, 71, 81, 99, 161, 165] [3, 33, 82, 92, 113, 166] [6, 39, 83, 96, 110, 167]
Code ID 168-14-10 · download JSON · raw on GitHub