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[[162,4,13]] d ≤
n
162
k
4
d
13
kd²/n
4.173
w
5
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 13, d_Z ≤ 13 · 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 13 · witness weight 13 (claimed upper_bound)
witness operator (support, 13 qubits)
[8, 33, 46, 69, 92, 99, 101, 105, 112, 114, 118, 125, 131]
d_Z 13 · witness weight 13 (claimed upper_bound)
witness operator (support, 13 qubits)
[9, 11, 18, 20, 26, 31, 32, 39, 52, 90, 104, 135, 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 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)×81 (2,2)×81 (3,2)×81 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 81 (1,3): 81 (2,2): 81 (2,3): 486 (2,4): 243 (3,2): 81 (3,3): 1539 (3,4): 2673 (3,5): 972 (3,6): 486 (3,7): 81
trapping sets H_Z (1,2)×81 (2,2)×81 (3,2)×81 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 81 (1,3): 81 (2,2): 81 (2,3): 486 (2,4): 243 (3,2): 81 (3,3): 1539 (3,4): 2673 (3,5): 972 (3,6): 486 (3,7): 81

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(81,6), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt5wtL2_order81_k4.txt, line 7. nonidentity supports a=[2], b=[7, 32].
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

[[162,4,13]] — two-block group-algebra code on SmallGroup(81,6)

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 = 4.17.

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_order81_k4.txt, line 7. This row is SmallGroup(81,6) with nonidentity GAP supports a=[2], b=[7, 32]. 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 ≤ 13, not an exact certificate. The published distance (d=13) 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(81,6), 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], b=[7, 32] 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) = 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 81 (max weight 5) · Z-checks 81 (max weight 5)
H_X (81 checks, sparse supports)
[0, 71, 81, 120, 147] [0, 1, 82, 106, 133] [2, 78, 83, 152, 153] [3, 5, 84, 101, 132] [4, 34, 85, 141, 157] [1, 5, 86, 92, 119] [2, 6, 81, 87, 157] [3, 7, 88, 114, 126] [4, 8, 89, 127, 150] [9, 80, 90, 143, 150] [10, 15, 86, 91, 140] [11, 52, 92, 115, 160] [12, 16, 93, 121, 148] [13, 17, 94, 122, 149] [14, 55, 95, 155, 161] [6, 15, 82, 96, 127] [7, 16, 97, 108, 139] [8, 17, 98, 109, 140] [9, 18, 99, 109, 152] [10, 19, 84, 100, 149] [11, 20, 85, 101, 161] [12, 21, 102, 134, 144] [13, 22, 103, 135, 145] [14, 23, 90, 104, 159] [24, 31, 82, 105, 135] [25, 66, 102, 106, 159] [26, 32, 97, 107, 154] [27, 33, 98, 99, 108] [28, 69, 109, 136, 138] [29, 35, 110, 142, 158] [30, 36, 87, 111, 112] [18, 31, 81, 112, 122] [19, 32, 88, 113, 145] [20, 33, 89, 90, 114] [21, 34, 115, 128, 153] [22, 35, 116, 129, 154] [23, 36, 83, 99, 117] [24, 37, 86, 118, 129] [25, 38, 83, 115, 119] [26, 39, 93, 120, 158] [27, 40, 94, 112, 121] [28, 41, 95, 122, 147] [29, 42, 123, 151, 156] [30, 43, 96, 105, 124] [44, 50, 88, 125, 151] [45, 51, 89, 96, 126] [46, 76, 123, 127, 133] [47, 53, 116, 118, 128] [48, 54, 117, 119, 129] [49, 56, 100, 130, 131] [37, 50, 84, 131, 142] [38, 51, 85, 87, 132] [39, 52, 102, 133, 156] [40, 53, 103, 105, 134] [41, 54, 104, 106, 135] [42, 55, 136, 146, 160] [43, 56, 91, 118, 137] [44, 57, 97, 138, 146] [45, 58, 91, 98, 139] [46, 59, 92, 136, 140] [47, 60, 110, 131, 141] [48, 61, 111, 132, 142] [49, 62, 113, 125, 143] [63, 67, 103, 113, 144] [64, 68, 104, 114, 145] [65, 70, 137, 139, 146] [57, 66, 93, 147, 155] [58, 67, 94, 100, 148] [59, 68, 95, 101, 149] [60, 69, 123, 125, 150] [61, 70, 124, 126, 151] [62, 71, 107, 138, 152] [63, 72, 107, 116, 153] [64, 73, 108, 117, 154] [65, 74, 130, 148, 155] [75, 77, 124, 134, 156] [72, 76, 110, 120, 157] [73, 77, 111, 121, 158] [74, 78, 143, 144, 159] [75, 79, 128, 137, 160] [79, 80, 130, 141, 161]
H_Z (81 checks, sparse supports)
[0, 6, 31, 81, 82] [1, 15, 24, 82, 86] [2, 36, 38, 83, 87] [3, 19, 50, 84, 88] [4, 20, 51, 85, 89] [5, 10, 37, 84, 86] [6, 30, 51, 87, 96] [7, 32, 44, 88, 97] [8, 33, 45, 89, 98] [9, 23, 33, 90, 99] [10, 56, 58, 91, 100] [5, 11, 59, 92, 101] [12, 39, 66, 93, 102] [13, 40, 67, 94, 103] [14, 41, 68, 95, 104] [15, 43, 45, 91, 96] [16, 26, 57, 93, 97] [17, 27, 58, 94, 98] [18, 27, 36, 99, 112] [19, 49, 67, 100, 113] [3, 20, 68, 101, 114] [21, 25, 52, 102, 115] [22, 53, 63, 103, 116] [23, 54, 64, 104, 117] [24, 43, 53, 105, 118] [1, 25, 54, 106, 119] [26, 71, 72, 107, 120] [16, 27, 73, 108, 121] [17, 18, 28, 109, 122] [29, 60, 76, 110, 123] [30, 61, 77, 111, 124] [30, 31, 40, 105, 112] [32, 62, 63, 107, 113] [7, 33, 64, 108, 114] [11, 34, 38, 85, 115] [35, 47, 72, 110, 116] [36, 48, 73, 111, 117] [37, 47, 56, 118, 131] [5, 38, 48, 119, 132] [0, 39, 76, 120, 133] [12, 40, 77, 121, 134] [13, 31, 41, 122, 135] [42, 46, 69, 123, 136] [43, 70, 75, 124, 137] [44, 62, 69, 125, 138] [7, 45, 70, 126, 139] [8, 15, 46, 127, 140] [34, 47, 79, 128, 141] [35, 37, 48, 129, 142] [49, 74, 80, 130, 143] [49, 50, 60, 125, 131] [3, 51, 61, 126, 132] [1, 46, 52, 92, 133] [21, 53, 75, 128, 134] [22, 24, 54, 129, 135] [28, 55, 59, 95, 136] [56, 65, 79, 130, 137] [28, 57, 71, 138, 147] [16, 58, 65, 139, 148] [10, 17, 59, 140, 149] [4, 60, 80, 141, 150] [29, 50, 61, 142, 151] [9, 62, 78, 143, 152] [21, 63, 78, 144, 153] [22, 32, 64, 145, 154] [55, 57, 65, 146, 155] [0, 41, 66, 106, 147] [12, 67, 74, 144, 148] [13, 19, 68, 145, 149] [8, 9, 69, 109, 150] [42, 44, 70, 146, 151] [2, 18, 71, 81, 152] [2, 34, 72, 153, 157] [26, 35, 73, 154, 158] [14, 66, 74, 155, 159] [42, 52, 75, 156, 160] [4, 6, 76, 127, 157] [29, 39, 77, 156, 158] [23, 25, 78, 83, 159] [11, 55, 79, 160, 161] [14, 20, 80, 90, 161]
Code ID 162-4-13 · download JSON · raw on GitHub