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[[176,4,20]] d ≤
n
176
k
4
d
20
kd²/n
9.091
w
8
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 20, d_Z ≤ 20 · w_X = 8, w_Z = 8 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[5, 19, 24, 39, 43, 60, 99, 107, 115, 119, 129, 133, 134, 137, 139, 146, 149, 167, 171, 175]
d_Z 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[25, 26, 31, 37, 41, 45, 50, 51, 54, 61, 65, 68, 70, 71, 83, 113, 121, 124, 152, 166]
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 8 · H_Z 8
qubit degrees H_X 2–6 (mean 4.0) · H_Z 2–6 (mean 4.0)
trapping sets H_X (1,2)×88 (2,2)×88 (3,2)×88 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,2): 88 (1,6): 88 (2,2): 88 (2,6): 1056 (2,10): 1320 (3,2): 88 (3,6): 6336 (3,8): 1408 (3,10): 26928 (3,12): 6248 (3,14): 22176 (3,16): 1760
trapping sets H_Z (1,2)×88 (2,2)×88 (3,2)×88 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,2): 88 (1,6): 88 (2,2): 88 (2,6): 1056 (2,10): 1320 (3,2): 88 (3,6): 6336 (3,8): 1408 (3,10): 26928 (3,12): 6248 (3,14): 22176 (3,16): 1760

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(88,3), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL2_order88_k4.txt, line 18. nonidentity supports a=[10], b=[2, 3, 5, 29, 50].
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 ≤ 8 (computed)

How this code was found

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

[[176,4,20]] — two-block group-algebra code on SmallGroup(88,3)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published weight-8 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 9.09.

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_wt8wtL2_order88_k4.txt, line 18. This row is SmallGroup(88,3) with nonidentity GAP supports a=[10], b=[2, 3, 5, 29, 50]. 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 ≤ 20, not an exact certificate. The published distance 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(88,3), 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=[10], b=[2, 3, 5, 29, 50] 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 = 176 - 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 88 (max weight 8) · Z-checks 88 (max weight 8)
H_X (88 checks, sparse supports)
[0, 86, 88, 94, 96, 109, 136, 163] [1, 13, 88, 89, 100, 112, 140, 167] [2, 75, 88, 90, 93, 103, 123, 169] [3, 81, 89, 90, 91, 116, 129, 170] [4, 8, 88, 92, 101, 104, 144, 173] [5, 14, 89, 93, 96, 106, 127, 172] [6, 20, 91, 93, 94, 105, 133, 173] [7, 21, 89, 95, 104, 108, 148, 163] [8, 82, 91, 96, 100, 110, 130, 174] [0, 9, 90, 92, 95, 97, 131, 172] [2, 10, 91, 97, 98, 108, 137, 167] [11, 16, 92, 93, 99, 112, 152, 166] [7, 12, 90, 94, 99, 100, 134, 175] [13, 22, 93, 95, 98, 101, 135, 174] [14, 28, 94, 97, 101, 102, 141, 170] [15, 29, 95, 96, 103, 116, 155, 156] [3, 16, 96, 98, 102, 104, 138, 175] [4, 17, 89, 97, 99, 105, 139, 165] [9, 18, 98, 100, 105, 106, 145, 159] [19, 24, 99, 107, 120, 158, 160, 172] [15, 20, 92, 100, 102, 108, 142, 169] [21, 30, 91, 101, 103, 109, 143, 168] [22, 36, 90, 102, 109, 110, 149, 162] [23, 37, 103, 111, 124, 147, 164, 174] [10, 24, 94, 104, 106, 112, 146, 171] [11, 25, 105, 107, 113, 147, 157, 167] [17, 26, 106, 113, 114, 151, 153, 175] [27, 32, 107, 115, 128, 150, 165, 168] [23, 28, 88, 108, 110, 116, 150, 161] [29, 38, 109, 111, 117, 151, 160, 170] [30, 44, 110, 117, 118, 154, 157, 169] [31, 45, 111, 119, 132, 139, 168, 171] [18, 32, 112, 114, 120, 154, 164, 173] [19, 33, 113, 115, 121, 149, 155, 159] [25, 34, 114, 121, 122, 143, 161, 171] [35, 40, 115, 123, 136, 142, 157, 174] [31, 36, 116, 118, 124, 153, 158, 163] [37, 46, 117, 119, 125, 152, 159, 162] [38, 52, 118, 125, 126, 146, 161, 165] [39, 53, 119, 127, 131, 140, 160, 175] [26, 40, 120, 122, 128, 156, 162, 166] [27, 41, 121, 123, 129, 141, 151, 163] [33, 42, 122, 129, 130, 135, 164, 169] [43, 48, 96, 123, 131, 134, 144, 149] [39, 44, 124, 126, 132, 145, 155, 166] [45, 54, 125, 127, 133, 144, 154, 167] [46, 60, 126, 133, 134, 138, 153, 172] [47, 61, 100, 123, 127, 135, 148, 152] [34, 48, 128, 130, 136, 148, 158, 170] [35, 49, 88, 129, 131, 133, 137, 143] [41, 50, 90, 127, 130, 137, 138, 156] [51, 56, 104, 126, 131, 139, 141, 152] [47, 52, 132, 134, 137, 140, 147, 173] [53, 62, 89, 133, 135, 136, 141, 146] [54, 68, 93, 130, 134, 141, 142, 145] [55, 69, 108, 115, 135, 143, 144, 156] [42, 56, 91, 136, 138, 140, 144, 150] [43, 57, 92, 125, 135, 137, 139, 145] [49, 58, 97, 119, 138, 145, 146, 148] [59, 64, 112, 118, 133, 139, 147, 160] [55, 60, 94, 129, 139, 140, 142, 148] [61, 70, 95, 128, 138, 141, 143, 149] [62, 76, 101, 122, 137, 142, 149, 150] [63, 77, 107, 116, 136, 143, 151, 164] [50, 64, 98, 132, 142, 144, 146, 152] [51, 65, 99, 117, 127, 145, 147, 153] [57, 66, 105, 111, 140, 146, 153, 154] [67, 72, 110, 120, 125, 147, 155, 168] [63, 68, 102, 121, 131, 148, 150, 156] [69, 78, 103, 120, 130, 149, 151, 157] [70, 83, 109, 114, 129, 150, 157, 158] [71, 84, 99, 124, 128, 151, 159, 171] [58, 72, 106, 124, 134, 152, 154, 160] [59, 73, 107, 109, 119, 153, 155, 161] [65, 74, 103, 113, 132, 154, 161, 162] [75, 80, 102, 117, 128, 155, 163, 174] [71, 76, 110, 113, 123, 156, 158, 164] [77, 85, 111, 112, 122, 157, 159, 165] [78, 87, 106, 117, 121, 158, 165, 166] [5, 79, 92, 120, 132, 159, 167, 175] [66, 80, 114, 116, 126, 160, 162, 168] [67, 81, 101, 111, 115, 161, 163, 169] [73, 82, 95, 121, 124, 162, 169, 170] [79, 83, 105, 115, 118, 164, 166, 171] [6, 84, 104, 114, 119, 165, 167, 172] [12, 85, 98, 113, 125, 166, 172, 173] [74, 86, 108, 118, 122, 168, 170, 174] [1, 87, 97, 107, 126, 171, 173, 175]
H_Z (88 checks, sparse supports)
[0, 1, 2, 4, 28, 49, 88, 97] [1, 3, 5, 7, 17, 53, 89, 175] [2, 3, 9, 12, 22, 50, 90, 98] [3, 6, 8, 10, 21, 56, 91, 104] [4, 9, 11, 20, 57, 79, 92, 105] [2, 5, 6, 11, 13, 54, 93, 167] [0, 6, 12, 14, 24, 60, 94, 172] [7, 9, 13, 15, 61, 82, 95, 100] [0, 5, 8, 15, 16, 43, 92, 96] [9, 10, 14, 17, 58, 87, 97, 106] [10, 13, 16, 18, 64, 85, 98, 112] [11, 12, 17, 19, 65, 71, 99, 113] [1, 8, 12, 18, 20, 47, 100, 173] [4, 13, 14, 21, 62, 81, 89, 101] [14, 16, 20, 22, 68, 75, 93, 102] [2, 15, 21, 23, 69, 74, 103, 108] [4, 7, 16, 24, 51, 84, 99, 104] [6, 17, 18, 25, 66, 83, 105, 114] [5, 18, 24, 26, 72, 78, 106, 120] [19, 25, 27, 63, 73, 87, 107, 121] [7, 10, 20, 28, 55, 86, 94, 108] [0, 21, 22, 29, 70, 73, 95, 109] [8, 22, 28, 30, 67, 76, 101, 110] [23, 29, 31, 66, 77, 81, 111, 116] [1, 11, 24, 32, 59, 77, 107, 112] [25, 26, 33, 74, 76, 85, 113, 122] [26, 32, 34, 70, 80, 84, 114, 128] [27, 33, 35, 55, 81, 83, 115, 129] [3, 15, 28, 36, 63, 80, 102, 116] [29, 30, 37, 65, 75, 78, 103, 117] [30, 36, 38, 59, 83, 86, 109, 118] [31, 37, 39, 58, 73, 84, 119, 124] [19, 32, 40, 67, 69, 79, 115, 120] [33, 34, 41, 68, 78, 82, 121, 130] [34, 40, 42, 62, 77, 86, 122, 136] [2, 35, 41, 43, 47, 76, 123, 137] [23, 36, 44, 71, 72, 82, 110, 124] [37, 38, 45, 57, 67, 85, 111, 125] [38, 44, 46, 51, 80, 87, 117, 126] [5, 39, 45, 47, 50, 65, 127, 132] [27, 40, 48, 61, 71, 75, 123, 128] [3, 41, 42, 49, 60, 70, 129, 138] [8, 42, 48, 50, 54, 69, 130, 144] [9, 39, 43, 49, 51, 68, 131, 145] [31, 44, 52, 64, 74, 79, 118, 132] [6, 45, 46, 49, 53, 59, 119, 133] [12, 43, 46, 52, 54, 72, 125, 134] [13, 42, 47, 53, 55, 57, 135, 140] [0, 35, 48, 53, 56, 63, 131, 136] [10, 49, 50, 52, 57, 62, 137, 146] [16, 46, 50, 56, 58, 61, 138, 152] [17, 31, 51, 57, 59, 60, 139, 153] [1, 39, 52, 56, 60, 66, 126, 140] [14, 41, 51, 53, 54, 61, 127, 141] [20, 35, 54, 60, 62, 64, 133, 142] [21, 34, 49, 55, 61, 63, 143, 148] [4, 43, 45, 55, 56, 64, 139, 144] [18, 44, 54, 57, 58, 65, 145, 154] [24, 38, 53, 58, 64, 66, 146, 160] [23, 25, 52, 59, 65, 67, 147, 161] [7, 47, 48, 58, 60, 68, 134, 148] [22, 33, 43, 61, 62, 69, 135, 149] [27, 28, 56, 62, 68, 70, 141, 150] [26, 29, 41, 63, 69, 71, 151, 156] [11, 37, 47, 51, 64, 72, 147, 152] [26, 36, 46, 65, 66, 73, 153, 162] [30, 32, 45, 66, 72, 74, 154, 168] [15, 33, 44, 67, 73, 75, 155, 169] [15, 40, 50, 55, 68, 76, 142, 156] [25, 30, 35, 69, 70, 77, 143, 157] [19, 36, 48, 70, 76, 78, 149, 158] [18, 33, 37, 71, 77, 79, 159, 164] [19, 29, 39, 59, 72, 80, 155, 160] [28, 34, 38, 73, 74, 81, 161, 170] [22, 37, 40, 74, 80, 82, 162, 174] [0, 7, 36, 41, 75, 81, 90, 163] [23, 32, 42, 63, 76, 83, 150, 164] [17, 27, 38, 77, 78, 84, 151, 165] [11, 40, 44, 78, 83, 85, 157, 166] [1, 10, 25, 45, 79, 84, 167, 171] [21, 27, 31, 67, 80, 86, 163, 168] [2, 20, 30, 42, 81, 82, 91, 169] [3, 14, 29, 48, 82, 86, 96, 170] [24, 31, 34, 71, 83, 87, 158, 171] [5, 9, 19, 46, 84, 85, 159, 172] [4, 6, 32, 52, 85, 87, 165, 173] [8, 13, 23, 35, 75, 86, 88, 174] [12, 16, 26, 39, 79, 87, 166, 175]
Code ID 176-4-20 · download JSON · raw on GitHub