← back to the board
[[184,2,22]] d ≤
n
184
k
2
d
22
kd²/n
5.261
w
8
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 22, d_Z ≤ 22 · 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 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[4, 15, 19, 34, 53, 57, 79, 86, 109, 113, 114, 122, 130, 137, 138, 142, 144, 150, 167, 172, 176, 182]
d_Z 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[21, 23, 26, 28, 29, 36, 51, 59, 67, 68, 71, 89, 109, 112, 118, 120, 124, 147, 152, 161, 181, 182]
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 4 · H_Z 4 (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 (2,4)×92 (3,4)×184 (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 (2,4): 92 (2,6): 2392 (3,4): 184 (3,6): 3864 (3,8): 42320 (3,10): 4140
trapping sets H_Z (1,4)×184 (2,4)×92 (3,4)×184 (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 (2,4): 92 (2,6): 2392 (3,4): 184 (3,6): 3864 (3,8): 42320 (3,10): 4140

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_k2.txt, line 11. nonidentity supports a=[2, 4, 8], b=[7, 36, 59].
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

[[184,2,22]] — 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 = 5.26.

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_k2.txt, line 11. This row is SmallGroup(92,1) with nonidentity GAP supports a=[2, 4, 8], b=[7, 36, 59]. 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=2), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 22, 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, 8], b=[7, 36, 59] 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) = 2. 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, 83, 87, 92, 130, 147, 182] [0, 1, 5, 9, 93, 132, 149, 183] [1, 2, 86, 90, 94, 127, 150, 179] [0, 3, 8, 87, 94, 95, 134, 151] [2, 4, 8, 12, 96, 129, 152, 181] [3, 5, 9, 13, 96, 97, 136, 153] [2, 5, 6, 90, 92, 98, 131, 154] [0, 3, 7, 12, 98, 99, 138, 155] [6, 8, 12, 16, 93, 100, 133, 156] [7, 9, 13, 17, 100, 101, 140, 157] [2, 6, 9, 10, 95, 102, 135, 158] [3, 7, 11, 16, 102, 103, 142, 159] [10, 12, 16, 20, 97, 104, 137, 160] [11, 13, 17, 21, 104, 105, 144, 161] [6, 10, 13, 14, 99, 106, 139, 162] [7, 11, 15, 20, 106, 107, 146, 163] [14, 16, 20, 24, 101, 108, 141, 164] [15, 17, 21, 25, 108, 109, 148, 165] [10, 14, 17, 18, 103, 110, 143, 166] [11, 15, 19, 24, 110, 111, 150, 167] [18, 20, 24, 28, 105, 112, 145, 168] [19, 21, 25, 29, 112, 113, 152, 169] [14, 18, 21, 22, 107, 114, 147, 170] [15, 19, 23, 28, 114, 115, 154, 171] [22, 24, 28, 32, 109, 116, 149, 172] [23, 25, 29, 33, 116, 117, 156, 173] [18, 22, 25, 26, 111, 118, 151, 174] [19, 23, 27, 32, 118, 119, 158, 175] [26, 28, 32, 36, 113, 120, 153, 176] [27, 29, 33, 37, 120, 121, 160, 177] [22, 26, 29, 30, 115, 122, 155, 178] [23, 27, 31, 36, 122, 123, 162, 179] [30, 32, 36, 40, 117, 124, 157, 180] [31, 33, 37, 41, 124, 125, 164, 181] [26, 30, 33, 34, 119, 126, 159, 182] [27, 31, 35, 40, 92, 126, 127, 166] [34, 36, 40, 44, 121, 128, 161, 183] [35, 37, 41, 45, 93, 128, 129, 168] [30, 34, 37, 38, 94, 123, 130, 163] [31, 35, 39, 44, 95, 130, 131, 170] [38, 40, 44, 48, 96, 125, 132, 165] [39, 41, 45, 49, 97, 132, 133, 172] [34, 38, 41, 42, 98, 127, 134, 167] [35, 39, 43, 48, 99, 134, 135, 174] [42, 44, 48, 52, 100, 129, 136, 169] [43, 45, 49, 53, 101, 136, 137, 176] [38, 42, 45, 46, 102, 131, 138, 171] [39, 43, 47, 52, 103, 138, 139, 178] [46, 48, 52, 56, 104, 133, 140, 173] [47, 49, 53, 57, 105, 140, 141, 180] [42, 46, 49, 50, 106, 135, 142, 175] [43, 47, 51, 56, 107, 142, 143, 182] [50, 52, 56, 60, 108, 137, 144, 177] [51, 53, 57, 61, 109, 144, 145, 183] [46, 50, 53, 54, 110, 139, 146, 179] [47, 51, 55, 60, 94, 111, 146, 147] [54, 56, 60, 64, 112, 141, 148, 181] [55, 57, 61, 65, 96, 113, 148, 149] [50, 54, 57, 58, 92, 114, 143, 150] [51, 55, 59, 64, 98, 115, 150, 151] [58, 60, 64, 68, 93, 116, 145, 152] [59, 61, 65, 69, 100, 117, 152, 153] [54, 58, 61, 62, 95, 118, 147, 154] [55, 59, 63, 68, 102, 119, 154, 155] [62, 64, 68, 72, 97, 120, 149, 156] [63, 65, 69, 73, 104, 121, 156, 157] [58, 62, 65, 66, 99, 122, 151, 158] [59, 63, 67, 72, 106, 123, 158, 159] [66, 68, 72, 76, 101, 124, 153, 160] [67, 69, 73, 77, 108, 125, 160, 161] [62, 66, 69, 70, 103, 126, 155, 162] [63, 67, 71, 76, 110, 127, 162, 163] [70, 72, 76, 80, 105, 128, 157, 164] [71, 73, 77, 81, 112, 129, 164, 165] [66, 70, 73, 74, 107, 130, 159, 166] [67, 71, 75, 80, 114, 131, 166, 167] [74, 76, 80, 84, 109, 132, 161, 168] [75, 77, 81, 85, 116, 133, 168, 169] [70, 74, 77, 78, 111, 134, 163, 170] [71, 75, 79, 84, 118, 135, 170, 171] [78, 80, 84, 88, 113, 136, 165, 172] [79, 81, 85, 89, 120, 137, 172, 173] [74, 78, 81, 82, 115, 138, 167, 174] [75, 79, 83, 88, 122, 139, 174, 175] [82, 84, 88, 91, 117, 140, 169, 176] [1, 83, 85, 89, 124, 141, 176, 177] [78, 82, 85, 86, 119, 142, 171, 178] [79, 83, 87, 91, 126, 143, 178, 179] [4, 86, 88, 91, 121, 144, 173, 180] [1, 5, 87, 89, 128, 145, 180, 181] [82, 86, 89, 90, 123, 146, 175, 182] [4, 8, 90, 91, 125, 148, 177, 183]
H_Z (92 checks, sparse supports)
[0, 6, 35, 58, 92, 93, 95, 99] [1, 8, 37, 60, 93, 94, 177, 181] [2, 3, 38, 55, 94, 96, 98, 102] [3, 10, 39, 62, 95, 97, 99, 103] [4, 5, 40, 57, 92, 96, 180, 183] [5, 12, 41, 64, 93, 97, 98, 181] [6, 7, 42, 59, 98, 100, 102, 106] [7, 14, 43, 66, 99, 101, 103, 107] [8, 9, 44, 61, 95, 96, 100, 183] [9, 16, 45, 68, 93, 97, 101, 102] [10, 11, 46, 63, 102, 104, 106, 110] [11, 18, 47, 70, 103, 105, 107, 111] [12, 13, 48, 65, 96, 99, 100, 104] [13, 20, 49, 72, 97, 101, 105, 106] [14, 15, 50, 67, 106, 108, 110, 114] [15, 22, 51, 74, 107, 109, 111, 115] [16, 17, 52, 69, 100, 103, 104, 108] [17, 24, 53, 76, 101, 105, 109, 110] [18, 19, 54, 71, 110, 112, 114, 118] [19, 26, 55, 78, 111, 113, 115, 119] [20, 21, 56, 73, 104, 107, 108, 112] [21, 28, 57, 80, 105, 109, 113, 114] [22, 23, 58, 75, 114, 116, 118, 122] [23, 30, 59, 82, 115, 117, 119, 123] [24, 25, 60, 77, 108, 111, 112, 116] [25, 32, 61, 84, 109, 113, 117, 118] [26, 27, 62, 79, 118, 120, 122, 126] [27, 34, 63, 86, 119, 121, 123, 127] [28, 29, 64, 81, 112, 115, 116, 120] [29, 36, 65, 88, 113, 117, 121, 122] [30, 31, 66, 83, 122, 124, 126, 130] [31, 38, 67, 90, 123, 125, 127, 131] [32, 33, 68, 85, 116, 119, 120, 124] [33, 40, 69, 91, 117, 121, 125, 126] [34, 35, 70, 87, 126, 128, 130, 134] [2, 35, 42, 71, 127, 129, 131, 135] [36, 37, 72, 89, 120, 123, 124, 128] [4, 37, 44, 73, 121, 125, 129, 130] [0, 38, 39, 74, 130, 132, 134, 138] [6, 39, 46, 75, 131, 133, 135, 139] [1, 40, 41, 76, 124, 127, 128, 132] [8, 41, 48, 77, 125, 129, 133, 134] [3, 42, 43, 78, 134, 136, 138, 142] [10, 43, 50, 79, 135, 137, 139, 143] [5, 44, 45, 80, 128, 131, 132, 136] [12, 45, 52, 81, 129, 133, 137, 138] [7, 46, 47, 82, 138, 140, 142, 146] [14, 47, 54, 83, 139, 141, 143, 147] [9, 48, 49, 84, 132, 135, 136, 140] [16, 49, 56, 85, 133, 137, 141, 142] [11, 50, 51, 86, 142, 144, 146, 150] [18, 51, 58, 87, 143, 145, 147, 151] [13, 52, 53, 88, 136, 139, 140, 144] [20, 53, 60, 89, 137, 141, 145, 146] [15, 54, 55, 90, 146, 148, 150, 154] [0, 22, 55, 62, 147, 149, 151, 155] [17, 56, 57, 91, 140, 143, 144, 148] [1, 24, 57, 64, 141, 145, 149, 150] [2, 19, 58, 59, 150, 152, 154, 158] [3, 26, 59, 66, 151, 153, 155, 159] [4, 21, 60, 61, 144, 147, 148, 152] [5, 28, 61, 68, 145, 149, 153, 154] [6, 23, 62, 63, 154, 156, 158, 162] [7, 30, 63, 70, 155, 157, 159, 163] [8, 25, 64, 65, 148, 151, 152, 156] [9, 32, 65, 72, 149, 153, 157, 158] [10, 27, 66, 67, 158, 160, 162, 166] [11, 34, 67, 74, 159, 161, 163, 167] [12, 29, 68, 69, 152, 155, 156, 160] [13, 36, 69, 76, 153, 157, 161, 162] [14, 31, 70, 71, 162, 164, 166, 170] [15, 38, 71, 78, 163, 165, 167, 171] [16, 33, 72, 73, 156, 159, 160, 164] [17, 40, 73, 80, 157, 161, 165, 166] [18, 35, 74, 75, 166, 168, 170, 174] [19, 42, 75, 82, 167, 169, 171, 175] [20, 37, 76, 77, 160, 163, 164, 168] [21, 44, 77, 84, 161, 165, 169, 170] [22, 39, 78, 79, 170, 172, 174, 178] [23, 46, 79, 86, 171, 173, 175, 179] [24, 41, 80, 81, 164, 167, 168, 172] [25, 48, 81, 88, 165, 169, 173, 174] [26, 43, 82, 83, 174, 176, 178, 182] [27, 50, 83, 90, 92, 175, 177, 179] [28, 45, 84, 85, 168, 171, 172, 176] [29, 52, 85, 91, 169, 173, 177, 178] [30, 47, 86, 87, 94, 178, 180, 182] [2, 31, 54, 87, 92, 95, 179, 181] [32, 49, 88, 89, 172, 175, 176, 180] [4, 33, 56, 89, 173, 177, 181, 182] [0, 34, 51, 90, 94, 98, 182, 183] [1, 36, 53, 91, 176, 179, 180, 183]
Code ID 184-2-22 · download JSON · raw on GitHub