← back to the board
[[188,4,21]] d ≤
n
188
k
4
d
21
kd²/n
9.383
w
8
X/Z
1

Share this result

Distance

X/Z asymmetry 1 · d_X ≤ 21, d_Z ≤ 21 · 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 21 · witness weight 21 (claimed upper_bound)
witness operator (support, 21 qubits)
[9, 10, 13, 24, 25, 30, 40, 52, 57, 87, 90, 105, 117, 131, 133, 146, 148, 155, 163, 178, 185]
d_Z 21 · witness weight 21 (claimed upper_bound)
witness operator (support, 21 qubits)
[7, 18, 25, 27, 49, 50, 61, 63, 66, 69, 74, 97, 98, 110, 118, 141, 142, 145, 156, 161, 170]
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)×188 (2,4)×282 (3,4)×188 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 188 (2,4): 282 (2,6): 2068 (3,4): 188 (3,6): 7426 (3,8): 33934 (3,10): 2632
trapping sets H_Z (1,4)×188 (2,4)×282 (3,4)×188 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 188 (2,4): 282 (2,6): 2068 (3,4): 188 (3,6): 7426 (3,8): 33934 (3,10): 2632

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(94,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order94_k4.txt, line 14. nonidentity supports a=[2, 3, 10], b=[3, 17, 51].
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

[[188,4,21]] — two-block group-algebra code on SmallGroup(94,1)

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 = 9.38.

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_order94_k4.txt, line 14. This row is SmallGroup(94,1) with nonidentity GAP supports a=[2, 3, 10], b=[3, 17, 51]. 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 ≤ 21, 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(94,1), 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, 10], b=[3, 17, 51] 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 = 188 - 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 94 (max weight 8) · Z-checks 94 (max weight 8)
H_X (94 checks, sparse supports)
[0, 1, 9, 92, 94, 138, 172, 186] [0, 1, 3, 86, 95, 139, 173, 187] [0, 2, 3, 11, 94, 96, 140, 174] [2, 3, 5, 88, 95, 97, 141, 175] [2, 4, 5, 13, 96, 98, 142, 176] [4, 5, 7, 90, 97, 99, 143, 177] [4, 6, 7, 15, 98, 100, 144, 178] [6, 7, 9, 92, 99, 101, 145, 179] [6, 8, 9, 17, 100, 102, 146, 180] [0, 8, 9, 11, 101, 103, 147, 181] [8, 10, 11, 19, 102, 104, 148, 182] [2, 10, 11, 13, 103, 105, 149, 183] [10, 12, 13, 21, 104, 106, 150, 184] [4, 12, 13, 15, 105, 107, 151, 185] [12, 14, 15, 23, 106, 108, 152, 186] [6, 14, 15, 17, 107, 109, 153, 187] [14, 16, 17, 25, 94, 108, 110, 154] [8, 16, 17, 19, 95, 109, 111, 155] [16, 18, 19, 27, 96, 110, 112, 156] [10, 18, 19, 21, 97, 111, 113, 157] [18, 20, 21, 29, 98, 112, 114, 158] [12, 20, 21, 23, 99, 113, 115, 159] [20, 22, 23, 31, 100, 114, 116, 160] [14, 22, 23, 25, 101, 115, 117, 161] [22, 24, 25, 33, 102, 116, 118, 162] [16, 24, 25, 27, 103, 117, 119, 163] [24, 26, 27, 35, 104, 118, 120, 164] [18, 26, 27, 29, 105, 119, 121, 165] [26, 28, 29, 37, 106, 120, 122, 166] [20, 28, 29, 31, 107, 121, 123, 167] [28, 30, 31, 39, 108, 122, 124, 168] [22, 30, 31, 33, 109, 123, 125, 169] [30, 32, 33, 41, 110, 124, 126, 170] [24, 32, 33, 35, 111, 125, 127, 171] [32, 34, 35, 43, 112, 126, 128, 172] [26, 34, 35, 37, 113, 127, 129, 173] [34, 36, 37, 45, 114, 128, 130, 174] [28, 36, 37, 39, 115, 129, 131, 175] [36, 38, 39, 47, 116, 130, 132, 176] [30, 38, 39, 41, 117, 131, 133, 177] [38, 40, 41, 49, 118, 132, 134, 178] [32, 40, 41, 43, 119, 133, 135, 179] [40, 42, 43, 51, 120, 134, 136, 180] [34, 42, 43, 45, 121, 135, 137, 181] [42, 44, 45, 53, 122, 136, 138, 182] [36, 44, 45, 47, 123, 137, 139, 183] [44, 46, 47, 55, 124, 138, 140, 184] [38, 46, 47, 49, 125, 139, 141, 185] [46, 48, 49, 57, 126, 140, 142, 186] [40, 48, 49, 51, 127, 141, 143, 187] [48, 50, 51, 59, 94, 128, 142, 144] [42, 50, 51, 53, 95, 129, 143, 145] [50, 52, 53, 61, 96, 130, 144, 146] [44, 52, 53, 55, 97, 131, 145, 147] [52, 54, 55, 63, 98, 132, 146, 148] [46, 54, 55, 57, 99, 133, 147, 149] [54, 56, 57, 65, 100, 134, 148, 150] [48, 56, 57, 59, 101, 135, 149, 151] [56, 58, 59, 67, 102, 136, 150, 152] [50, 58, 59, 61, 103, 137, 151, 153] [58, 60, 61, 69, 104, 138, 152, 154] [52, 60, 61, 63, 105, 139, 153, 155] [60, 62, 63, 71, 106, 140, 154, 156] [54, 62, 63, 65, 107, 141, 155, 157] [62, 64, 65, 73, 108, 142, 156, 158] [56, 64, 65, 67, 109, 143, 157, 159] [64, 66, 67, 75, 110, 144, 158, 160] [58, 66, 67, 69, 111, 145, 159, 161] [66, 68, 69, 77, 112, 146, 160, 162] [60, 68, 69, 71, 113, 147, 161, 163] [68, 70, 71, 79, 114, 148, 162, 164] [62, 70, 71, 73, 115, 149, 163, 165] [70, 72, 73, 81, 116, 150, 164, 166] [64, 72, 73, 75, 117, 151, 165, 167] [72, 74, 75, 83, 118, 152, 166, 168] [66, 74, 75, 77, 119, 153, 167, 169] [74, 76, 77, 85, 120, 154, 168, 170] [68, 76, 77, 79, 121, 155, 169, 171] [76, 78, 79, 87, 122, 156, 170, 172] [70, 78, 79, 81, 123, 157, 171, 173] [78, 80, 81, 89, 124, 158, 172, 174] [72, 80, 81, 83, 125, 159, 173, 175] [80, 82, 83, 91, 126, 160, 174, 176] [74, 82, 83, 85, 127, 161, 175, 177] [82, 84, 85, 93, 128, 162, 176, 178] [76, 84, 85, 87, 129, 163, 177, 179] [1, 84, 86, 87, 130, 164, 178, 180] [78, 86, 87, 89, 131, 165, 179, 181] [3, 86, 88, 89, 132, 166, 180, 182] [80, 88, 89, 91, 133, 167, 181, 183] [5, 88, 90, 91, 134, 168, 182, 184] [82, 90, 91, 93, 135, 169, 183, 185] [7, 90, 92, 93, 136, 170, 184, 186] [1, 84, 92, 93, 137, 171, 185, 187]
H_Z (94 checks, sparse supports)
[0, 2, 16, 50, 94, 95, 96, 103] [1, 3, 17, 51, 94, 95, 180, 187] [2, 4, 18, 52, 96, 97, 98, 105] [3, 5, 19, 53, 95, 96, 97, 182] [4, 6, 20, 54, 98, 99, 100, 107] [5, 7, 21, 55, 97, 98, 99, 184] [6, 8, 22, 56, 100, 101, 102, 109] [7, 9, 23, 57, 99, 100, 101, 186] [8, 10, 24, 58, 102, 103, 104, 111] [9, 11, 25, 59, 94, 101, 102, 103] [10, 12, 26, 60, 104, 105, 106, 113] [11, 13, 27, 61, 96, 103, 104, 105] [12, 14, 28, 62, 106, 107, 108, 115] [13, 15, 29, 63, 98, 105, 106, 107] [14, 16, 30, 64, 108, 109, 110, 117] [15, 17, 31, 65, 100, 107, 108, 109] [16, 18, 32, 66, 110, 111, 112, 119] [17, 19, 33, 67, 102, 109, 110, 111] [18, 20, 34, 68, 112, 113, 114, 121] [19, 21, 35, 69, 104, 111, 112, 113] [20, 22, 36, 70, 114, 115, 116, 123] [21, 23, 37, 71, 106, 113, 114, 115] [22, 24, 38, 72, 116, 117, 118, 125] [23, 25, 39, 73, 108, 115, 116, 117] [24, 26, 40, 74, 118, 119, 120, 127] [25, 27, 41, 75, 110, 117, 118, 119] [26, 28, 42, 76, 120, 121, 122, 129] [27, 29, 43, 77, 112, 119, 120, 121] [28, 30, 44, 78, 122, 123, 124, 131] [29, 31, 45, 79, 114, 121, 122, 123] [30, 32, 46, 80, 124, 125, 126, 133] [31, 33, 47, 81, 116, 123, 124, 125] [32, 34, 48, 82, 126, 127, 128, 135] [33, 35, 49, 83, 118, 125, 126, 127] [34, 36, 50, 84, 128, 129, 130, 137] [35, 37, 51, 85, 120, 127, 128, 129] [36, 38, 52, 86, 130, 131, 132, 139] [37, 39, 53, 87, 122, 129, 130, 131] [38, 40, 54, 88, 132, 133, 134, 141] [39, 41, 55, 89, 124, 131, 132, 133] [40, 42, 56, 90, 134, 135, 136, 143] [41, 43, 57, 91, 126, 133, 134, 135] [42, 44, 58, 92, 136, 137, 138, 145] [43, 45, 59, 93, 128, 135, 136, 137] [0, 44, 46, 60, 138, 139, 140, 147] [1, 45, 47, 61, 130, 137, 138, 139] [2, 46, 48, 62, 140, 141, 142, 149] [3, 47, 49, 63, 132, 139, 140, 141] [4, 48, 50, 64, 142, 143, 144, 151] [5, 49, 51, 65, 134, 141, 142, 143] [6, 50, 52, 66, 144, 145, 146, 153] [7, 51, 53, 67, 136, 143, 144, 145] [8, 52, 54, 68, 146, 147, 148, 155] [9, 53, 55, 69, 138, 145, 146, 147] [10, 54, 56, 70, 148, 149, 150, 157] [11, 55, 57, 71, 140, 147, 148, 149] [12, 56, 58, 72, 150, 151, 152, 159] [13, 57, 59, 73, 142, 149, 150, 151] [14, 58, 60, 74, 152, 153, 154, 161] [15, 59, 61, 75, 144, 151, 152, 153] [16, 60, 62, 76, 154, 155, 156, 163] [17, 61, 63, 77, 146, 153, 154, 155] [18, 62, 64, 78, 156, 157, 158, 165] [19, 63, 65, 79, 148, 155, 156, 157] [20, 64, 66, 80, 158, 159, 160, 167] [21, 65, 67, 81, 150, 157, 158, 159] [22, 66, 68, 82, 160, 161, 162, 169] [23, 67, 69, 83, 152, 159, 160, 161] [24, 68, 70, 84, 162, 163, 164, 171] [25, 69, 71, 85, 154, 161, 162, 163] [26, 70, 72, 86, 164, 165, 166, 173] [27, 71, 73, 87, 156, 163, 164, 165] [28, 72, 74, 88, 166, 167, 168, 175] [29, 73, 75, 89, 158, 165, 166, 167] [30, 74, 76, 90, 168, 169, 170, 177] [31, 75, 77, 91, 160, 167, 168, 169] [32, 76, 78, 92, 170, 171, 172, 179] [33, 77, 79, 93, 162, 169, 170, 171] [0, 34, 78, 80, 172, 173, 174, 181] [1, 35, 79, 81, 164, 171, 172, 173] [2, 36, 80, 82, 174, 175, 176, 183] [3, 37, 81, 83, 166, 173, 174, 175] [4, 38, 82, 84, 176, 177, 178, 185] [5, 39, 83, 85, 168, 175, 176, 177] [6, 40, 84, 86, 178, 179, 180, 187] [7, 41, 85, 87, 170, 177, 178, 179] [8, 42, 86, 88, 95, 180, 181, 182] [9, 43, 87, 89, 172, 179, 180, 181] [10, 44, 88, 90, 97, 182, 183, 184] [11, 45, 89, 91, 174, 181, 182, 183] [12, 46, 90, 92, 99, 184, 185, 186] [13, 47, 91, 93, 176, 183, 184, 185] [0, 14, 48, 92, 94, 101, 186, 187] [1, 15, 49, 93, 178, 185, 186, 187]
Code ID 188-4-21 · download JSON · raw on GitHub