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[[184,4,20]] d ≤
n
184
k
4
d
20
kd²/n
8.696
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)
[0, 5, 78, 82, 87, 88, 93, 125, 129, 133, 137, 140, 144, 148, 152, 165, 169, 173, 177, 181]
d_Z 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[8, 26, 30, 32, 40, 86, 90, 91, 99, 110, 119, 127, 142, 143, 151, 154, 159, 163, 171, 175]
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 4 · H_Z 4
trapping sets H_X (1,4)×184 (2,6)×2576 (3,6)×1748 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 184 (2,6): 2576 (3,6): 1748 (3,8): 48852 (3,10): 5152
trapping sets H_Z (1,4)×184 (2,6)×2576 (3,6)×1748 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 184 (2,6): 2576 (3,6): 1748 (3,8): 48852 (3,10): 5152

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(92,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order92_k4.txt, line 12. nonidentity supports a=[2, 4, 10], b=[15, 39, 55].
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

[[184,4,20]] — two-block group-algebra code on SmallGroup(92,1)

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 = 8.70.

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_wt8wtL4_order92_k4.txt, line 12. This row is SmallGroup(92,1) with nonidentity GAP supports a=[2, 4, 10], b=[15, 39, 55]. 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(92,1), 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, 4, 10], b=[15, 39, 55] 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 = 184 - 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 92 (max weight 8) · Z-checks 92 (max weight 8)
H_X (92 checks, sparse supports)
[0, 4, 12, 87, 92, 134, 150, 174] [0, 1, 5, 83, 93, 136, 152, 176] [1, 2, 9, 90, 94, 131, 147, 171] [0, 3, 8, 16, 95, 138, 154, 178] [2, 4, 8, 86, 96, 133, 149, 173] [3, 5, 9, 87, 97, 140, 156, 180] [2, 5, 6, 13, 98, 135, 151, 175] [3, 7, 12, 20, 99, 142, 158, 182] [6, 8, 12, 90, 100, 137, 153, 177] [0, 7, 9, 13, 101, 144, 160, 183] [6, 9, 10, 17, 102, 139, 155, 179] [7, 11, 16, 24, 94, 103, 146, 162] [2, 10, 12, 16, 104, 141, 157, 181] [3, 11, 13, 17, 96, 105, 148, 164] [10, 13, 14, 21, 92, 106, 143, 159] [11, 15, 20, 28, 98, 107, 150, 166] [6, 14, 16, 20, 93, 108, 145, 161] [7, 15, 17, 21, 100, 109, 152, 168] [14, 17, 18, 25, 95, 110, 147, 163] [15, 19, 24, 32, 102, 111, 154, 170] [10, 18, 20, 24, 97, 112, 149, 165] [11, 19, 21, 25, 104, 113, 156, 172] [18, 21, 22, 29, 99, 114, 151, 167] [19, 23, 28, 36, 106, 115, 158, 174] [14, 22, 24, 28, 101, 116, 153, 169] [15, 23, 25, 29, 108, 117, 160, 176] [22, 25, 26, 33, 103, 118, 155, 171] [23, 27, 32, 40, 110, 119, 162, 178] [18, 26, 28, 32, 105, 120, 157, 173] [19, 27, 29, 33, 112, 121, 164, 180] [26, 29, 30, 37, 107, 122, 159, 175] [27, 31, 36, 44, 114, 123, 166, 182] [22, 30, 32, 36, 109, 124, 161, 177] [23, 31, 33, 37, 116, 125, 168, 183] [30, 33, 34, 41, 111, 126, 163, 179] [31, 35, 40, 48, 94, 118, 127, 170] [26, 34, 36, 40, 113, 128, 165, 181] [27, 35, 37, 41, 96, 120, 129, 172] [34, 37, 38, 45, 92, 115, 130, 167] [35, 39, 44, 52, 98, 122, 131, 174] [30, 38, 40, 44, 93, 117, 132, 169] [31, 39, 41, 45, 100, 124, 133, 176] [38, 41, 42, 49, 95, 119, 134, 171] [39, 43, 48, 56, 102, 126, 135, 178] [34, 42, 44, 48, 97, 121, 136, 173] [35, 43, 45, 49, 104, 128, 137, 180] [42, 45, 46, 53, 99, 123, 138, 175] [43, 47, 52, 60, 106, 130, 139, 182] [38, 46, 48, 52, 101, 125, 140, 177] [39, 47, 49, 53, 108, 132, 141, 183] [46, 49, 50, 57, 103, 127, 142, 179] [47, 51, 56, 64, 94, 110, 134, 143] [42, 50, 52, 56, 105, 129, 144, 181] [43, 51, 53, 57, 96, 112, 136, 145] [50, 53, 54, 61, 92, 107, 131, 146] [51, 55, 60, 68, 98, 114, 138, 147] [46, 54, 56, 60, 93, 109, 133, 148] [47, 55, 57, 61, 100, 116, 140, 149] [54, 57, 58, 65, 95, 111, 135, 150] [55, 59, 64, 72, 102, 118, 142, 151] [50, 58, 60, 64, 97, 113, 137, 152] [51, 59, 61, 65, 104, 120, 144, 153] [58, 61, 62, 69, 99, 115, 139, 154] [59, 63, 68, 76, 106, 122, 146, 155] [54, 62, 64, 68, 101, 117, 141, 156] [55, 63, 65, 69, 108, 124, 148, 157] [62, 65, 66, 73, 103, 119, 143, 158] [63, 67, 72, 80, 110, 126, 150, 159] [58, 66, 68, 72, 105, 121, 145, 160] [59, 67, 69, 73, 112, 128, 152, 161] [66, 69, 70, 77, 107, 123, 147, 162] [67, 71, 76, 84, 114, 130, 154, 163] [62, 70, 72, 76, 109, 125, 149, 164] [63, 71, 73, 77, 116, 132, 156, 165] [70, 73, 74, 81, 111, 127, 151, 166] [71, 75, 80, 88, 118, 134, 158, 167] [66, 74, 76, 80, 113, 129, 153, 168] [67, 75, 77, 81, 120, 136, 160, 169] [74, 77, 78, 85, 115, 131, 155, 170] [75, 79, 84, 91, 122, 138, 162, 171] [70, 78, 80, 84, 117, 133, 157, 172] [71, 79, 81, 85, 124, 140, 164, 173] [78, 81, 82, 89, 119, 135, 159, 174] [4, 79, 83, 88, 126, 142, 166, 175] [74, 82, 84, 88, 121, 137, 161, 176] [75, 83, 85, 89, 128, 144, 168, 177] [1, 82, 85, 86, 123, 139, 163, 178] [8, 83, 87, 91, 130, 146, 170, 179] [78, 86, 88, 91, 125, 141, 165, 180] [1, 79, 87, 89, 132, 148, 172, 181] [5, 86, 89, 90, 127, 143, 167, 182] [4, 82, 90, 91, 129, 145, 169, 183]
H_Z (92 checks, sparse supports)
[0, 14, 38, 54, 92, 93, 95, 101] [1, 16, 40, 56, 93, 94, 178, 181] [2, 11, 35, 51, 94, 96, 98, 104] [3, 18, 42, 58, 95, 97, 99, 105] [4, 13, 37, 53, 92, 96, 175, 183] [5, 20, 44, 60, 93, 97, 98, 182] [6, 15, 39, 55, 98, 100, 102, 108] [7, 22, 46, 62, 99, 101, 103, 109] [8, 17, 41, 57, 95, 96, 100, 179] [9, 24, 48, 64, 94, 97, 101, 102] [10, 19, 43, 59, 102, 104, 106, 112] [11, 26, 50, 66, 103, 105, 107, 113] [12, 21, 45, 61, 92, 99, 100, 104] [13, 28, 52, 68, 98, 101, 105, 106] [14, 23, 47, 63, 106, 108, 110, 116] [15, 30, 54, 70, 107, 109, 111, 117] [16, 25, 49, 65, 95, 103, 104, 108] [17, 32, 56, 72, 102, 105, 109, 110] [18, 27, 51, 67, 110, 112, 114, 120] [19, 34, 58, 74, 111, 113, 115, 121] [20, 29, 53, 69, 99, 107, 108, 112] [21, 36, 60, 76, 106, 109, 113, 114] [22, 31, 55, 71, 114, 116, 118, 124] [23, 38, 62, 78, 115, 117, 119, 125] [24, 33, 57, 73, 103, 111, 112, 116] [25, 40, 64, 80, 110, 113, 117, 118] [26, 35, 59, 75, 118, 120, 122, 128] [27, 42, 66, 82, 119, 121, 123, 129] [28, 37, 61, 77, 107, 115, 116, 120] [29, 44, 68, 84, 114, 117, 121, 122] [30, 39, 63, 79, 122, 124, 126, 132] [31, 46, 70, 86, 123, 125, 127, 133] [32, 41, 65, 81, 111, 119, 120, 124] [33, 48, 72, 88, 118, 121, 125, 126] [34, 43, 67, 83, 126, 128, 130, 136] [35, 50, 74, 90, 127, 129, 131, 137] [36, 45, 69, 85, 115, 123, 124, 128] [37, 52, 76, 91, 122, 125, 129, 130] [38, 47, 71, 87, 130, 132, 134, 140] [2, 39, 54, 78, 131, 133, 135, 141] [40, 49, 73, 89, 119, 127, 128, 132] [4, 41, 56, 80, 126, 129, 133, 134] [0, 42, 51, 75, 134, 136, 138, 144] [6, 43, 58, 82, 135, 137, 139, 145] [1, 44, 53, 77, 123, 131, 132, 136] [8, 45, 60, 84, 130, 133, 137, 138] [3, 46, 55, 79, 138, 140, 142, 148] [10, 47, 62, 86, 139, 141, 143, 149] [5, 48, 57, 81, 127, 135, 136, 140] [12, 49, 64, 88, 134, 137, 141, 142] [7, 50, 59, 83, 142, 144, 146, 152] [14, 51, 66, 90, 143, 145, 147, 153] [9, 52, 61, 85, 131, 139, 140, 144] [16, 53, 68, 91, 138, 141, 145, 146] [11, 54, 63, 87, 146, 148, 150, 156] [2, 18, 55, 70, 147, 149, 151, 157] [13, 56, 65, 89, 135, 143, 144, 148] [4, 20, 57, 72, 142, 145, 149, 150] [0, 15, 58, 67, 150, 152, 154, 160] [6, 22, 59, 74, 151, 153, 155, 161] [1, 17, 60, 69, 139, 147, 148, 152] [8, 24, 61, 76, 146, 149, 153, 154] [3, 19, 62, 71, 154, 156, 158, 164] [10, 26, 63, 78, 155, 157, 159, 165] [5, 21, 64, 73, 143, 151, 152, 156] [12, 28, 65, 80, 150, 153, 157, 158] [7, 23, 66, 75, 158, 160, 162, 168] [14, 30, 67, 82, 159, 161, 163, 169] [9, 25, 68, 77, 147, 155, 156, 160] [16, 32, 69, 84, 154, 157, 161, 162] [11, 27, 70, 79, 162, 164, 166, 172] [18, 34, 71, 86, 163, 165, 167, 173] [13, 29, 72, 81, 151, 159, 160, 164] [20, 36, 73, 88, 158, 161, 165, 166] [15, 31, 74, 83, 166, 168, 170, 176] [22, 38, 75, 90, 167, 169, 171, 177] [17, 33, 76, 85, 155, 163, 164, 168] [24, 40, 77, 91, 162, 165, 169, 170] [19, 35, 78, 87, 170, 172, 174, 180] [2, 26, 42, 79, 171, 173, 175, 181] [21, 37, 80, 89, 159, 167, 168, 172] [4, 28, 44, 81, 166, 169, 173, 174] [0, 23, 39, 82, 174, 176, 178, 183] [6, 30, 46, 83, 93, 175, 177, 179] [1, 25, 41, 84, 163, 171, 172, 176] [8, 32, 48, 85, 170, 173, 177, 178] [3, 27, 43, 86, 96, 178, 180, 182] [10, 34, 50, 87, 92, 97, 179, 181] [5, 29, 45, 88, 167, 175, 176, 180] [12, 36, 52, 89, 174, 177, 181, 182] [7, 31, 47, 90, 94, 100, 182, 183] [9, 33, 49, 91, 171, 179, 180, 183]
Code ID 184-4-20 · download JSON · raw on GitHub