← back to the board
[[184,94,2]] d =
n
184
k
94
d
2
kd²/n
2.043
w
8
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X = 2, d_Z = 2 · 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 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[123, 128]
d_Z 2 · witness weight 2 (claimed upper_bound)
witness operator (support, 2 qubits)
[63, 65]
certificate exact, d = 2 · scipy/HiGHS MILP
X: no logical < 2 exists; Z: no logical < 2 exists

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 (2,0)×184 (3,4)×6532 (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 (2,0): 184 (2,4): 920 (3,4): 6532 (3,8): 736
trapping sets H_Z (1,4)×184 (2,0)×184 (3,4)×6532 (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 (2,0): 184 (2,4): 920 (3,4): 6532 (3,8): 736

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_k94.txt, line 1. nonidentity supports a=[2, 3, 5], b=[3, 4, 7].
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

[[184,94,2]] — 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 = 2.04.

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_k94.txt, line 1. This row is SmallGroup(92,1) with nonidentity GAP supports a=[2, 3, 5], b=[3, 4, 7]. 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=94), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 2, 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, 3, 5], b=[3, 4, 7] 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) = 94. 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, 1, 2, 4, 92, 94, 179, 182] [0, 1, 2, 4, 93, 96, 181, 183] [0, 1, 2, 4, 92, 94, 179, 182] [3, 5, 6, 8, 92, 94, 95, 98] [0, 1, 2, 4, 93, 96, 181, 183] [3, 5, 6, 8, 93, 96, 97, 100] [3, 5, 6, 8, 92, 94, 95, 98] [7, 9, 10, 12, 95, 98, 99, 102] [3, 5, 6, 8, 93, 96, 97, 100] [7, 9, 10, 12, 97, 100, 101, 104] [7, 9, 10, 12, 95, 98, 99, 102] [11, 13, 14, 16, 99, 102, 103, 106] [7, 9, 10, 12, 97, 100, 101, 104] [11, 13, 14, 16, 101, 104, 105, 108] [11, 13, 14, 16, 99, 102, 103, 106] [15, 17, 18, 20, 103, 106, 107, 110] [11, 13, 14, 16, 101, 104, 105, 108] [15, 17, 18, 20, 105, 108, 109, 112] [15, 17, 18, 20, 103, 106, 107, 110] [19, 21, 22, 24, 107, 110, 111, 114] [15, 17, 18, 20, 105, 108, 109, 112] [19, 21, 22, 24, 109, 112, 113, 116] [19, 21, 22, 24, 107, 110, 111, 114] [23, 25, 26, 28, 111, 114, 115, 118] [19, 21, 22, 24, 109, 112, 113, 116] [23, 25, 26, 28, 113, 116, 117, 120] [23, 25, 26, 28, 111, 114, 115, 118] [27, 29, 30, 32, 115, 118, 119, 122] [23, 25, 26, 28, 113, 116, 117, 120] [27, 29, 30, 32, 117, 120, 121, 124] [27, 29, 30, 32, 115, 118, 119, 122] [31, 33, 34, 36, 119, 122, 123, 126] [27, 29, 30, 32, 117, 120, 121, 124] [31, 33, 34, 36, 121, 124, 125, 128] [31, 33, 34, 36, 119, 122, 123, 126] [35, 37, 38, 40, 123, 126, 127, 130] [31, 33, 34, 36, 121, 124, 125, 128] [35, 37, 38, 40, 125, 128, 129, 132] [35, 37, 38, 40, 123, 126, 127, 130] [39, 41, 42, 44, 127, 130, 131, 134] [35, 37, 38, 40, 125, 128, 129, 132] [39, 41, 42, 44, 129, 132, 133, 136] [39, 41, 42, 44, 127, 130, 131, 134] [43, 45, 46, 48, 131, 134, 135, 138] [39, 41, 42, 44, 129, 132, 133, 136] [43, 45, 46, 48, 133, 136, 137, 140] [43, 45, 46, 48, 131, 134, 135, 138] [47, 49, 50, 52, 135, 138, 139, 142] [43, 45, 46, 48, 133, 136, 137, 140] [47, 49, 50, 52, 137, 140, 141, 144] [47, 49, 50, 52, 135, 138, 139, 142] [51, 53, 54, 56, 139, 142, 143, 146] [47, 49, 50, 52, 137, 140, 141, 144] [51, 53, 54, 56, 141, 144, 145, 148] [51, 53, 54, 56, 139, 142, 143, 146] [55, 57, 58, 60, 143, 146, 147, 150] [51, 53, 54, 56, 141, 144, 145, 148] [55, 57, 58, 60, 145, 148, 149, 152] [55, 57, 58, 60, 143, 146, 147, 150] [59, 61, 62, 64, 147, 150, 151, 154] [55, 57, 58, 60, 145, 148, 149, 152] [59, 61, 62, 64, 149, 152, 153, 156] [59, 61, 62, 64, 147, 150, 151, 154] [63, 65, 66, 68, 151, 154, 155, 158] [59, 61, 62, 64, 149, 152, 153, 156] [63, 65, 66, 68, 153, 156, 157, 160] [63, 65, 66, 68, 151, 154, 155, 158] [67, 69, 70, 72, 155, 158, 159, 162] [63, 65, 66, 68, 153, 156, 157, 160] [67, 69, 70, 72, 157, 160, 161, 164] [67, 69, 70, 72, 155, 158, 159, 162] [71, 73, 74, 76, 159, 162, 163, 166] [67, 69, 70, 72, 157, 160, 161, 164] [71, 73, 74, 76, 161, 164, 165, 168] [71, 73, 74, 76, 159, 162, 163, 166] [75, 77, 78, 80, 163, 166, 167, 170] [71, 73, 74, 76, 161, 164, 165, 168] [75, 77, 78, 80, 165, 168, 169, 172] [75, 77, 78, 80, 163, 166, 167, 170] [79, 81, 82, 84, 167, 170, 171, 174] [75, 77, 78, 80, 165, 168, 169, 172] [79, 81, 82, 84, 169, 172, 173, 176] [79, 81, 82, 84, 167, 170, 171, 174] [83, 85, 86, 88, 171, 174, 175, 178] [79, 81, 82, 84, 169, 172, 173, 176] [83, 85, 86, 88, 173, 176, 177, 180] [83, 85, 86, 88, 171, 174, 175, 178] [87, 89, 90, 91, 175, 178, 179, 182] [83, 85, 86, 88, 173, 176, 177, 180] [87, 89, 90, 91, 177, 180, 181, 183] [87, 89, 90, 91, 175, 178, 179, 182] [87, 89, 90, 91, 177, 180, 181, 183]
H_Z (92 checks, sparse supports)
[0, 2, 3, 6, 92, 93, 94, 96] [1, 4, 5, 8, 92, 93, 94, 96] [0, 2, 3, 6, 92, 93, 94, 96] [3, 6, 7, 10, 95, 97, 98, 100] [1, 4, 5, 8, 92, 93, 94, 96] [5, 8, 9, 12, 95, 97, 98, 100] [3, 6, 7, 10, 95, 97, 98, 100] [7, 10, 11, 14, 99, 101, 102, 104] [5, 8, 9, 12, 95, 97, 98, 100] [9, 12, 13, 16, 99, 101, 102, 104] [7, 10, 11, 14, 99, 101, 102, 104] [11, 14, 15, 18, 103, 105, 106, 108] [9, 12, 13, 16, 99, 101, 102, 104] [13, 16, 17, 20, 103, 105, 106, 108] [11, 14, 15, 18, 103, 105, 106, 108] [15, 18, 19, 22, 107, 109, 110, 112] [13, 16, 17, 20, 103, 105, 106, 108] [17, 20, 21, 24, 107, 109, 110, 112] [15, 18, 19, 22, 107, 109, 110, 112] [19, 22, 23, 26, 111, 113, 114, 116] [17, 20, 21, 24, 107, 109, 110, 112] [21, 24, 25, 28, 111, 113, 114, 116] [19, 22, 23, 26, 111, 113, 114, 116] [23, 26, 27, 30, 115, 117, 118, 120] [21, 24, 25, 28, 111, 113, 114, 116] [25, 28, 29, 32, 115, 117, 118, 120] [23, 26, 27, 30, 115, 117, 118, 120] [27, 30, 31, 34, 119, 121, 122, 124] [25, 28, 29, 32, 115, 117, 118, 120] [29, 32, 33, 36, 119, 121, 122, 124] [27, 30, 31, 34, 119, 121, 122, 124] [31, 34, 35, 38, 123, 125, 126, 128] [29, 32, 33, 36, 119, 121, 122, 124] [33, 36, 37, 40, 123, 125, 126, 128] [31, 34, 35, 38, 123, 125, 126, 128] [35, 38, 39, 42, 127, 129, 130, 132] [33, 36, 37, 40, 123, 125, 126, 128] [37, 40, 41, 44, 127, 129, 130, 132] [35, 38, 39, 42, 127, 129, 130, 132] [39, 42, 43, 46, 131, 133, 134, 136] [37, 40, 41, 44, 127, 129, 130, 132] [41, 44, 45, 48, 131, 133, 134, 136] [39, 42, 43, 46, 131, 133, 134, 136] [43, 46, 47, 50, 135, 137, 138, 140] [41, 44, 45, 48, 131, 133, 134, 136] [45, 48, 49, 52, 135, 137, 138, 140] [43, 46, 47, 50, 135, 137, 138, 140] [47, 50, 51, 54, 139, 141, 142, 144] [45, 48, 49, 52, 135, 137, 138, 140] [49, 52, 53, 56, 139, 141, 142, 144] [47, 50, 51, 54, 139, 141, 142, 144] [51, 54, 55, 58, 143, 145, 146, 148] [49, 52, 53, 56, 139, 141, 142, 144] [53, 56, 57, 60, 143, 145, 146, 148] [51, 54, 55, 58, 143, 145, 146, 148] [55, 58, 59, 62, 147, 149, 150, 152] [53, 56, 57, 60, 143, 145, 146, 148] [57, 60, 61, 64, 147, 149, 150, 152] [55, 58, 59, 62, 147, 149, 150, 152] [59, 62, 63, 66, 151, 153, 154, 156] [57, 60, 61, 64, 147, 149, 150, 152] [61, 64, 65, 68, 151, 153, 154, 156] [59, 62, 63, 66, 151, 153, 154, 156] [63, 66, 67, 70, 155, 157, 158, 160] [61, 64, 65, 68, 151, 153, 154, 156] [65, 68, 69, 72, 155, 157, 158, 160] [63, 66, 67, 70, 155, 157, 158, 160] [67, 70, 71, 74, 159, 161, 162, 164] [65, 68, 69, 72, 155, 157, 158, 160] [69, 72, 73, 76, 159, 161, 162, 164] [67, 70, 71, 74, 159, 161, 162, 164] [71, 74, 75, 78, 163, 165, 166, 168] [69, 72, 73, 76, 159, 161, 162, 164] [73, 76, 77, 80, 163, 165, 166, 168] [71, 74, 75, 78, 163, 165, 166, 168] [75, 78, 79, 82, 167, 169, 170, 172] [73, 76, 77, 80, 163, 165, 166, 168] [77, 80, 81, 84, 167, 169, 170, 172] [75, 78, 79, 82, 167, 169, 170, 172] [79, 82, 83, 86, 171, 173, 174, 176] [77, 80, 81, 84, 167, 169, 170, 172] [81, 84, 85, 88, 171, 173, 174, 176] [79, 82, 83, 86, 171, 173, 174, 176] [83, 86, 87, 90, 175, 177, 178, 180] [81, 84, 85, 88, 171, 173, 174, 176] [85, 88, 89, 91, 175, 177, 178, 180] [83, 86, 87, 90, 175, 177, 178, 180] [0, 2, 87, 90, 179, 181, 182, 183] [85, 88, 89, 91, 175, 177, 178, 180] [1, 4, 89, 91, 179, 181, 182, 183] [0, 2, 87, 90, 179, 181, 182, 183] [1, 4, 89, 91, 179, 181, 182, 183]
Code ID 184-94-2 · download JSON · raw on GitHub