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[[162,16,9]] d ≤
n
162
k
16
d
9
kd²/n
8.0
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 9, d_Z ≤ 9 · 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 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[9, 29, 48, 99, 112, 123, 136, 142, 151]
d_Z 9 · witness weight 9 (claimed upper_bound)
witness operator (support, 9 qubits)
[26, 57, 73, 77, 93, 100, 102, 119, 138]
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 (3,4)×270 (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 (3,4): 270 (3,5): 972 (3,6): 7938 (3,7): 12717 (3,8): 3888 (3,9): 1458 (3,10): 324
trapping sets H_Z (1,3)×81 (2,4)×243 (3,4)×270 (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 (3,4): 270 (3,5): 972 (3,6): 7938 (3,7): 12717 (3,8): 3888 (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

[[162,16,9]] — 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 = 8.00.

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_k16.txt, line 3. This row is SmallGroup(81,3) with nonidentity GAP supports a=[2, 16], b=[3, 14, 45]. 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 ≤ 9, not an exact certificate. The published distance (d=9) 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, 16] and b=[3, 14, 45] 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) = 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 81 (max weight 7) · Z-checks 81 (max weight 7)
H_X (81 checks, sparse supports)
[0, 36, 79, 81, 90, 91, 130] [0, 1, 77, 82, 99, 100, 143] [2, 54, 62, 81, 83, 105, 146] [3, 56, 59, 84, 95, 105, 107] [4, 5, 57, 85, 93, 106, 108] [1, 5, 9, 86, 112, 113, 152] [2, 6, 56, 82, 87, 118, 155] [3, 7, 80, 88, 104, 118, 120] [4, 8, 50, 89, 102, 119, 121] [9, 68, 74, 83, 84, 90, 156] [10, 23, 70, 84, 91, 109, 125] [11, 15, 21, 85, 92, 107, 126] [12, 71, 73, 83, 93, 111, 125] [13, 16, 18, 81, 94, 126, 128] [14, 17, 72, 95, 110, 127, 129] [0, 6, 15, 86, 96, 131, 159] [7, 16, 24, 97, 117, 131, 133] [8, 17, 25, 98, 115, 132, 134] [9, 18, 70, 87, 88, 99, 160] [10, 19, 71, 88, 100, 122, 138] [11, 16, 20, 89, 101, 120, 139] [12, 21, 68, 87, 102, 124, 138] [13, 22, 66, 82, 103, 139, 141] [14, 23, 67, 104, 123, 140, 142] [24, 41, 77, 91, 93, 105, 127] [25, 31, 39, 92, 94, 106, 125] [26, 43, 78, 90, 93, 107, 129] [1, 27, 32, 83, 94, 108, 144] [28, 33, 42, 95, 109, 128, 145] [29, 34, 37, 84, 92, 110, 144] [30, 35, 38, 85, 111, 145, 146] [2, 18, 31, 96, 97, 112, 161] [3, 19, 32, 97, 113, 135, 147] [4, 20, 33, 98, 114, 133, 148] [21, 34, 44, 96, 115, 137, 147] [22, 35, 45, 86, 116, 148, 150] [23, 36, 46, 117, 136, 149, 151] [24, 37, 78, 100, 102, 118, 140] [25, 32, 38, 101, 103, 119, 138] [26, 36, 39, 99, 102, 120, 142] [27, 34, 40, 87, 103, 121, 153] [28, 35, 41, 104, 122, 141, 154] [29, 31, 42, 88, 101, 123, 153] [30, 43, 76, 89, 124, 154, 155] [44, 61, 80, 81, 107, 125, 145] [6, 45, 50, 90, 108, 110, 126] [46, 51, 60, 109, 111, 127, 144] [7, 47, 52, 91, 106, 110, 128] [8, 48, 53, 92, 111, 129, 156] [49, 55, 58, 94, 109, 130, 156] [10, 37, 50, 113, 115, 131, 149] [11, 38, 51, 114, 116, 132, 147] [12, 39, 52, 112, 115, 133, 151] [13, 40, 53, 96, 116, 134, 157] [14, 41, 54, 117, 135, 150, 158] [42, 55, 63, 97, 114, 136, 157] [43, 56, 64, 98, 137, 158, 159] [44, 54, 57, 82, 120, 138, 154] [45, 52, 58, 99, 121, 123, 139] [46, 53, 59, 122, 124, 140, 153] [5, 47, 60, 100, 119, 123, 141] [48, 55, 61, 101, 124, 142, 160] [49, 51, 62, 103, 122, 143, 160] [19, 63, 66, 85, 105, 128, 144] [20, 64, 67, 106, 129, 130, 145] [22, 65, 69, 108, 127, 130, 146] [26, 57, 66, 86, 133, 147, 158] [27, 58, 67, 112, 134, 136, 148] [28, 59, 68, 135, 137, 149, 157] [29, 60, 69, 113, 132, 136, 150] [30, 61, 70, 114, 137, 151, 161] [62, 71, 75, 116, 135, 152, 161] [15, 63, 72, 89, 118, 141, 153] [64, 69, 73, 119, 142, 143, 154] [17, 65, 74, 121, 140, 143, 155] [40, 75, 76, 95, 126, 146, 156] [47, 72, 76, 98, 131, 150, 157] [48, 73, 77, 132, 151, 152, 158] [49, 74, 78, 134, 149, 152, 159] [33, 75, 79, 104, 139, 155, 160] [65, 79, 80, 117, 148, 159, 161]
H_Z (81 checks, sparse supports)
[0, 2, 13, 44, 81, 82, 96] [1, 6, 22, 57, 82, 86, 108] [2, 9, 12, 27, 83, 87, 112] [3, 9, 10, 29, 84, 88, 113] [4, 11, 30, 63, 85, 89, 114] [5, 15, 35, 66, 85, 86, 141] [6, 18, 21, 40, 87, 96, 126] [7, 18, 19, 42, 88, 97, 128] [8, 20, 43, 72, 89, 98, 129] [0, 9, 26, 45, 86, 90, 99] [0, 10, 24, 47, 91, 100, 131] [11, 25, 29, 48, 92, 101, 132] [4, 12, 24, 26, 93, 102, 133] [13, 25, 27, 49, 94, 103, 134] [3, 14, 28, 75, 95, 104, 135] [15, 31, 34, 53, 92, 96, 153] [16, 31, 32, 55, 94, 97, 101] [17, 33, 56, 76, 95, 98, 155] [1, 18, 39, 58, 94, 99, 112] [1, 19, 37, 60, 100, 113, 144] [20, 38, 42, 61, 101, 114, 145] [8, 21, 37, 39, 92, 102, 115] [22, 38, 40, 62, 103, 116, 146] [7, 23, 41, 79, 91, 104, 117] [2, 3, 24, 63, 97, 105, 118] [4, 25, 47, 64, 98, 106, 119] [3, 11, 26, 44, 107, 120, 147] [4, 27, 45, 65, 108, 121, 148] [10, 28, 46, 49, 109, 122, 149] [14, 29, 45, 47, 110, 123, 150] [12, 30, 46, 48, 111, 124, 151] [5, 31, 52, 67, 106, 112, 123] [5, 32, 50, 69, 108, 113, 119] [33, 51, 55, 70, 109, 114, 160] [17, 34, 50, 52, 110, 115, 121] [35, 51, 53, 71, 111, 116, 122] [16, 36, 54, 80, 81, 117, 120] [6, 7, 37, 72, 110, 118, 131] [8, 38, 60, 73, 111, 119, 132] [7, 20, 39, 57, 106, 120, 133] [8, 40, 58, 74, 121, 134, 156] [19, 41, 59, 62, 105, 122, 135] [23, 42, 58, 60, 109, 123, 136] [21, 43, 59, 61, 107, 124, 137] [10, 12, 25, 44, 115, 125, 138] [11, 13, 45, 75, 116, 126, 139] [14, 24, 46, 65, 117, 127, 140] [13, 28, 47, 63, 128, 141, 157] [14, 26, 48, 64, 129, 142, 158] [0, 49, 64, 65, 130, 143, 159] [15, 16, 50, 76, 89, 126, 131] [17, 51, 69, 77, 127, 132, 143] [16, 33, 52, 66, 128, 133, 139] [17, 53, 67, 78, 129, 134, 140] [32, 54, 68, 71, 83, 135, 138] [36, 55, 67, 69, 130, 136, 142] [34, 56, 68, 70, 84, 87, 137] [19, 21, 38, 57, 85, 138, 147] [20, 22, 58, 79, 130, 139, 148] [23, 37, 59, 74, 84, 140, 149] [22, 41, 60, 72, 127, 141, 150] [23, 39, 61, 73, 125, 142, 151] [1, 62, 73, 74, 83, 143, 152] [27, 29, 46, 63, 136, 144, 153] [28, 30, 44, 64, 137, 145, 154] [2, 30, 65, 75, 146, 155, 161] [32, 34, 51, 66, 103, 144, 147] [33, 35, 67, 80, 104, 145, 148] [36, 50, 68, 78, 90, 102, 149] [35, 54, 69, 76, 146, 150, 154] [36, 52, 70, 77, 91, 99, 151] [5, 71, 77, 78, 93, 100, 152] [40, 42, 59, 72, 95, 153, 157] [41, 43, 57, 73, 93, 154, 158] [6, 43, 74, 79, 90, 155, 159] [9, 48, 49, 75, 152, 156, 160] [53, 55, 68, 76, 124, 156, 157] [54, 56, 66, 77, 82, 105, 158] [15, 56, 78, 80, 107, 118, 159] [18, 61, 62, 79, 81, 160, 161] [31, 70, 71, 80, 88, 125, 161]
Code ID 162-16-9 · download JSON · raw on GitHub