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[[162,8,15]] d ≤
n
162
k
8
d
15
kd²/n
11.111
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 15, d_Z ≤ 15 · 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 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[20, 29, 50, 52, 63, 64, 80, 91, 107, 111, 114, 120, 126, 133, 142]
d_Z 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[0, 16, 19, 30, 36, 43, 52, 71, 88, 108, 113, 122, 126, 140, 145]
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)×81 (2,4)×243 (3,3)×81 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 81 (1,4): 81 (2,4): 243 (2,5): 972 (2,6): 486 (3,3): 81 (3,4): 27 (3,5): 1161 (3,6): 8370 (3,7): 11421 (3,8): 4779 (3,9): 1458 (3,10): 324
trapping sets H_Z (1,3)×81 (2,4)×243 (3,3)×81 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 81 (1,4): 81 (2,4): 243 (2,5): 972 (2,6): 486 (3,3): 81 (3,4): 27 (3,5): 1161 (3,6): 8370 (3,7): 11421 (3,8): 4779 (3,9): 1458 (3,10): 324

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 ≤ 8 (computed)

How this code was found

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

[[162,8,15]] — two-block group-algebra code on SmallGroup(81,3)

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 = 11.11.

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_order81_k8.txt, line 6. This row is SmallGroup(81,3) with nonidentity GAP supports a=[2, 17], b=[3, 19, 52]. 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 ≤ 15, not an exact certificate. The published distance (d=15) 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(81,3), 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=[2, 17] and b=[3, 19, 52] 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 = 162 - 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 81 (max weight 7) · Z-checks 81 (max weight 7)
H_X (81 checks, sparse supports)
[0, 36, 62, 81, 90, 121, 159] [0, 1, 71, 82, 99, 107, 134] [2, 54, 74, 81, 83, 153, 158] [3, 23, 56, 84, 105, 135, 141] [4, 5, 21, 85, 106, 133, 142] [1, 5, 12, 86, 112, 120, 129] [2, 6, 78, 82, 87, 105, 157] [3, 7, 36, 83, 88, 118, 150] [4, 8, 34, 89, 119, 128, 151] [9, 68, 79, 83, 89, 90, 117] [10, 41, 70, 84, 91, 119, 161] [11, 15, 39, 85, 92, 131, 160] [12, 43, 71, 93, 101, 125, 151] [1, 13, 16, 94, 96, 126, 155] [14, 17, 42, 95, 100, 127, 150] [6, 15, 26, 86, 96, 118, 156] [0, 7, 16, 87, 97, 131, 146] [8, 17, 29, 91, 98, 132, 141] [9, 18, 80, 81, 87, 98, 99] [10, 19, 54, 88, 100, 125, 132] [11, 20, 52, 89, 101, 126, 161] [12, 21, 56, 91, 102, 114, 138] [5, 13, 22, 92, 103, 139, 159] [14, 23, 55, 104, 113, 140, 146] [24, 59, 77, 91, 103, 105, 137] [25, 31, 57, 86, 92, 104, 106] [26, 61, 78, 93, 107, 139, 149] [6, 27, 32, 94, 108, 140, 147] [28, 33, 60, 95, 109, 138, 148] [7, 29, 34, 110, 113, 122, 144] [8, 30, 35, 111, 114, 120, 145] [18, 31, 44, 82, 95, 96, 112] [2, 19, 32, 97, 113, 127, 138] [20, 33, 47, 98, 114, 125, 139] [3, 21, 34, 100, 109, 115, 147] [4, 22, 35, 101, 107, 116, 148] [23, 36, 49, 108, 117, 149, 155] [24, 37, 68, 84, 100, 116, 118] [25, 38, 66, 85, 101, 117, 119] [26, 39, 70, 90, 102, 120, 148] [15, 27, 40, 103, 121, 144, 149] [28, 41, 69, 104, 122, 145, 147] [16, 29, 42, 108, 123, 135, 153] [17, 30, 43, 109, 124, 133, 154] [44, 73, 80, 107, 123, 125, 152] [18, 45, 50, 97, 108, 124, 126] [46, 51, 72, 82, 98, 109, 127] [19, 47, 52, 110, 112, 128, 154] [20, 48, 53, 99, 111, 129, 157] [22, 49, 55, 87, 130, 134, 156] [9, 37, 50, 88, 111, 113, 131] [38, 51, 63, 81, 89, 114, 132] [10, 39, 52, 99, 115, 133, 145] [11, 40, 53, 90, 116, 134, 153] [41, 54, 65, 117, 135, 144, 154] [13, 42, 55, 83, 121, 136, 157] [14, 43, 56, 122, 128, 137, 158] [44, 57, 77, 93, 120, 136, 138] [31, 45, 58, 94, 121, 137, 139] [46, 59, 76, 86, 95, 122, 140] [32, 47, 60, 106, 123, 141, 158] [33, 48, 61, 112, 124, 142, 156] [35, 49, 62, 96, 129, 143, 160] [37, 63, 66, 115, 128, 143, 144] [38, 64, 67, 88, 116, 129, 145] [40, 65, 69, 118, 130, 132, 146] [24, 57, 66, 102, 130, 133, 147] [25, 58, 67, 84, 103, 134, 148] [59, 68, 75, 85, 104, 135, 149] [27, 60, 69, 105, 119, 136, 150] [28, 61, 70, 106, 137, 151, 160] [30, 62, 71, 92, 142, 152, 161] [50, 63, 72, 110, 141, 152, 153] [51, 64, 73, 97, 111, 142, 154] [53, 65, 74, 127, 131, 143, 155] [58, 75, 76, 102, 136, 146, 156] [45, 72, 76, 93, 123, 150, 157] [46, 73, 77, 94, 124, 151, 158] [48, 74, 78, 126, 140, 152, 159] [67, 75, 79, 115, 130, 155, 160] [64, 79, 80, 110, 143, 159, 161]
H_Z (81 checks, sparse supports)
[0, 2, 18, 51, 81, 82, 97] [1, 6, 31, 46, 82, 86, 94] [2, 7, 9, 55, 83, 87, 113] [3, 10, 37, 67, 84, 88, 115] [4, 11, 38, 68, 85, 89, 116] [5, 15, 25, 59, 85, 86, 103] [6, 16, 18, 49, 87, 96, 108] [7, 19, 50, 64, 88, 97, 110] [8, 9, 20, 51, 89, 98, 111] [0, 9, 39, 53, 90, 99, 131] [10, 17, 21, 24, 91, 100, 133] [11, 22, 25, 71, 92, 101, 134] [12, 26, 57, 76, 86, 93, 102] [13, 27, 58, 77, 94, 103, 136] [14, 28, 31, 59, 95, 104, 137] [13, 15, 31, 62, 92, 96, 121] [16, 32, 45, 73, 94, 97, 123] [17, 18, 33, 46, 95, 98, 124] [1, 18, 48, 52, 99, 112, 126] [14, 19, 34, 37, 100, 113, 128] [12, 20, 35, 38, 101, 114, 129] [21, 39, 66, 75, 85, 102, 115] [22, 24, 40, 67, 103, 116, 130] [23, 25, 41, 68, 84, 104, 117] [3, 6, 24, 69, 105, 118, 147] [4, 25, 60, 70, 106, 119, 148] [1, 26, 35, 44, 96, 107, 120] [27, 36, 42, 45, 108, 121, 150] [28, 34, 43, 46, 109, 122, 151] [29, 47, 72, 80, 98, 110, 123] [30, 48, 50, 73, 111, 124, 152] [5, 31, 47, 61, 106, 112, 139] [23, 29, 32, 50, 108, 113, 141] [21, 30, 33, 51, 109, 114, 142] [34, 52, 63, 79, 89, 110, 115] [35, 37, 53, 64, 111, 116, 143] [9, 36, 38, 54, 81, 88, 117] [7, 15, 37, 65, 118, 131, 144] [8, 10, 38, 69, 119, 132, 145] [5, 30, 39, 57, 92, 120, 133] [0, 40, 55, 58, 121, 134, 146] [29, 41, 56, 59, 91, 122, 135] [42, 44, 60, 76, 95, 123, 136] [43, 45, 61, 77, 93, 124, 137] [12, 19, 33, 44, 112, 125, 138] [13, 20, 45, 78, 126, 139, 157] [14, 32, 46, 74, 127, 140, 158] [8, 47, 56, 63, 114, 128, 141] [5, 48, 62, 64, 129, 142, 159] [49, 65, 66, 79, 117, 130, 143] [11, 16, 50, 74, 126, 131, 153] [17, 19, 51, 65, 127, 132, 154] [4, 43, 52, 66, 101, 128, 133] [1, 49, 53, 67, 129, 134, 155] [3, 42, 54, 68, 83, 100, 135] [55, 57, 69, 75, 104, 130, 136] [24, 56, 58, 70, 84, 102, 137] [21, 28, 32, 57, 106, 138, 147] [22, 26, 33, 58, 139, 148, 156] [23, 27, 59, 78, 105, 140, 149] [3, 17, 60, 72, 109, 141, 150] [4, 61, 71, 73, 107, 142, 151] [62, 63, 74, 80, 81, 143, 152] [29, 40, 54, 63, 132, 144, 153] [30, 41, 52, 64, 145, 154, 161] [16, 23, 65, 75, 135, 146, 155] [27, 34, 41, 66, 119, 144, 147] [28, 35, 39, 67, 145, 148, 160] [26, 36, 40, 68, 90, 118, 149] [7, 14, 69, 76, 122, 146, 150] [8, 12, 70, 77, 91, 120, 151] [44, 71, 72, 78, 82, 93, 152] [2, 42, 53, 72, 127, 153, 157] [43, 47, 54, 73, 125, 154, 158] [13, 36, 74, 79, 83, 155, 159] [15, 49, 61, 75, 149, 156, 160] [6, 48, 55, 76, 140, 156, 157] [2, 56, 60, 77, 105, 138, 158] [0, 22, 78, 80, 87, 107, 159] [11, 62, 70, 79, 90, 160, 161] [10, 20, 71, 80, 99, 125, 161]
Code ID 162-8-15 · download JSON · raw on GitHub