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[[196,4,21]] d ≤
n
196
k
4
d
21
kd²/n
9.0
w
8
X/Z
1.05

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Distance

X/Z asymmetry 1.05 · d_X ≤ 21, 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 21 · witness weight 21 (claimed upper_bound)
witness operator (support, 21 qubits)
[0, 3, 8, 15, 20, 24, 31, 35, 44, 48, 58, 70, 80, 88, 98, 101, 106, 113, 122, 133, 146]
d_Z 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[41, 44, 45, 56, 59, 63, 64, 66, 81, 83, 98, 107, 133, 139, 141, 144, 145, 161, 164, 170, 176, 195]
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)×196 (2,4)×294 (3,4)×196 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 196 (2,4): 294 (2,6): 2156 (3,4): 196 (3,6): 7742 (3,8): 35378 (3,10): 2744
trapping sets H_Z (1,4)×196 (2,4)×294 (3,4)×196 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 196 (2,4): 294 (2,6): 2156 (3,4): 196 (3,6): 7742 (3,8): 35378 (3,10): 2744

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on SmallGroup(98,1), from github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c, nonabelian.zip, diswtnonabelian_wt8wtL4_order98_k4.txt, line 10. nonidentity supports a=[2, 3, 6], b=[3, 9, 89].
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

[[196,4,21]] — two-block group-algebra code on SmallGroup(98,1)

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 = 9.00.

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_order98_k4.txt, line 10. This row is SmallGroup(98,1) with nonidentity GAP supports a=[2, 3, 6], b=[3, 9, 89]. 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(98,1), 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, 6], b=[3, 9, 89] 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 = 196 - 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 98 (max weight 8) · Z-checks 98 (max weight 8)
H_X (98 checks, sparse supports)
[0, 1, 5, 96, 98, 110, 133, 194] [0, 1, 4, 48, 99, 114, 139, 195] [0, 2, 4, 10, 98, 100, 118, 145] [3, 5, 11, 42, 101, 119, 140, 146] [2, 4, 9, 61, 99, 102, 123, 152] [0, 3, 5, 10, 103, 124, 147, 153] [2, 6, 9, 17, 100, 104, 128, 158] [3, 7, 10, 18, 101, 105, 129, 159] [8, 11, 19, 56, 98, 106, 130, 154] [6, 9, 16, 72, 102, 107, 134, 164] [2, 7, 10, 17, 103, 108, 135, 165] [3, 8, 11, 18, 99, 109, 136, 160] [6, 12, 16, 26, 104, 110, 140, 169] [7, 13, 17, 27, 105, 111, 141, 170] [8, 14, 18, 28, 100, 106, 112, 142] [15, 19, 29, 68, 101, 113, 143, 166] [12, 16, 25, 81, 107, 114, 147, 174] [6, 13, 17, 26, 108, 115, 148, 175] [7, 14, 18, 27, 102, 109, 116, 149] [8, 15, 19, 28, 103, 117, 150, 171] [12, 20, 25, 37, 101, 110, 118, 178] [13, 21, 26, 38, 111, 119, 154, 179] [14, 22, 27, 39, 104, 112, 120, 155] [15, 23, 28, 40, 105, 113, 121, 156] [24, 29, 41, 78, 106, 122, 157, 176] [20, 25, 36, 88, 103, 114, 123, 182] [12, 21, 26, 37, 115, 124, 160, 183] [13, 22, 27, 38, 107, 116, 125, 161] [14, 23, 28, 39, 108, 117, 126, 162] [15, 24, 29, 40, 109, 127, 163, 180] [20, 30, 36, 50, 105, 118, 128, 185] [21, 31, 37, 51, 106, 119, 129, 186] [22, 32, 38, 52, 110, 120, 130, 166] [23, 33, 39, 53, 111, 121, 131, 167] [24, 34, 40, 54, 112, 122, 132, 168] [35, 41, 55, 86, 113, 133, 169, 184] [30, 36, 49, 93, 108, 123, 134, 188] [20, 31, 37, 50, 109, 124, 135, 189] [21, 32, 38, 51, 114, 125, 136, 171] [22, 33, 39, 52, 115, 126, 137, 172] [23, 34, 40, 53, 116, 127, 138, 173] [24, 35, 41, 54, 117, 139, 174, 187] [30, 42, 49, 62, 111, 128, 140, 190] [31, 43, 50, 63, 112, 129, 141, 191] [32, 44, 51, 64, 113, 118, 130, 142] [33, 45, 52, 65, 119, 131, 143, 176] [34, 46, 53, 66, 120, 132, 144, 177] [35, 47, 54, 67, 121, 133, 145, 178] [1, 48, 55, 92, 122, 146, 179, 190] [5, 42, 49, 96, 115, 134, 147, 192] [30, 43, 50, 62, 116, 135, 148, 193] [31, 44, 51, 63, 117, 123, 136, 149] [32, 45, 52, 64, 124, 137, 150, 180] [33, 46, 53, 65, 125, 138, 151, 181] [34, 47, 54, 66, 126, 139, 152, 182] [35, 48, 55, 67, 127, 153, 183, 192] [43, 56, 62, 73, 120, 141, 154, 194] [44, 57, 63, 74, 121, 128, 142, 155] [45, 58, 64, 75, 122, 129, 143, 156] [46, 59, 65, 76, 130, 144, 157, 184] [47, 60, 66, 77, 131, 145, 158, 185] [4, 48, 61, 67, 132, 146, 159, 186] [11, 42, 56, 62, 125, 148, 160, 195] [43, 57, 63, 73, 126, 134, 149, 161] [44, 58, 64, 74, 127, 135, 150, 162] [45, 59, 65, 75, 136, 151, 163, 187] [46, 60, 66, 76, 137, 152, 164, 188] [47, 61, 67, 77, 138, 153, 165, 189] [57, 68, 73, 82, 131, 140, 155, 166] [58, 69, 74, 83, 132, 141, 156, 167] [59, 70, 75, 84, 133, 142, 157, 168] [60, 71, 76, 85, 143, 158, 169, 190] [9, 61, 72, 77, 144, 159, 170, 191] [19, 56, 68, 73, 137, 147, 161, 171] [57, 69, 74, 82, 138, 148, 162, 172] [58, 70, 75, 83, 139, 149, 163, 173] [59, 71, 76, 84, 150, 164, 174, 192] [60, 72, 77, 85, 151, 165, 175, 193] [69, 78, 82, 89, 144, 154, 167, 176] [70, 79, 83, 90, 145, 155, 168, 177] [71, 80, 84, 91, 146, 156, 169, 178] [16, 72, 81, 85, 157, 170, 179, 194] [29, 68, 78, 82, 151, 160, 172, 180] [69, 79, 83, 89, 152, 161, 173, 181] [70, 80, 84, 90, 153, 162, 174, 182] [71, 81, 85, 91, 163, 175, 183, 195] [79, 86, 89, 94, 158, 166, 177, 184] [80, 87, 90, 95, 159, 167, 178, 185] [25, 81, 88, 91, 98, 168, 179, 186] [41, 78, 86, 89, 164, 171, 181, 187] [79, 87, 90, 94, 165, 172, 182, 188] [80, 88, 91, 95, 99, 173, 183, 189] [87, 92, 94, 97, 170, 176, 185, 190] [36, 88, 93, 95, 100, 177, 186, 191] [55, 86, 92, 94, 175, 180, 188, 192] [87, 93, 95, 97, 102, 181, 189, 193] [49, 93, 96, 97, 104, 184, 191, 194] [1, 92, 96, 97, 107, 187, 193, 195]
H_Z (98 checks, sparse supports)
[0, 2, 8, 88, 98, 99, 100, 103] [1, 4, 11, 91, 98, 99, 146, 195] [2, 6, 14, 93, 100, 102, 104, 108] [3, 7, 15, 20, 101, 103, 105, 109] [4, 9, 18, 95, 99, 100, 102, 159] [5, 10, 19, 25, 98, 101, 103, 147] [6, 12, 22, 96, 104, 107, 110, 115] [7, 13, 23, 30, 105, 108, 111, 116] [8, 14, 24, 31, 106, 109, 112, 117] [9, 16, 27, 97, 102, 104, 107, 170] [10, 17, 28, 36, 100, 103, 105, 108] [11, 18, 29, 37, 101, 106, 109, 160] [0, 12, 20, 32, 110, 114, 118, 124] [13, 21, 33, 42, 111, 115, 119, 125] [14, 22, 34, 43, 112, 116, 120, 126] [15, 23, 35, 44, 113, 117, 121, 127] [1, 16, 25, 38, 107, 110, 114, 179] [17, 26, 39, 49, 104, 108, 111, 115] [18, 27, 40, 50, 105, 109, 112, 116] [19, 28, 41, 51, 106, 113, 117, 171] [2, 20, 30, 44, 118, 123, 128, 135] [3, 21, 31, 45, 119, 124, 129, 136] [22, 32, 46, 56, 120, 125, 130, 137] [23, 33, 47, 57, 121, 126, 131, 138] [24, 34, 48, 58, 122, 127, 132, 139] [4, 25, 36, 51, 114, 118, 123, 186] [5, 26, 37, 52, 110, 115, 119, 124] [27, 38, 53, 62, 111, 116, 120, 125] [28, 39, 54, 63, 112, 117, 121, 126] [29, 40, 55, 64, 113, 122, 127, 180] [6, 30, 42, 57, 128, 134, 140, 148] [7, 31, 43, 58, 129, 135, 141, 149] [8, 32, 44, 59, 130, 136, 142, 150] [33, 45, 60, 68, 131, 137, 143, 151] [34, 46, 61, 69, 132, 138, 144, 152] [0, 35, 47, 70, 133, 139, 145, 153] [9, 36, 49, 63, 123, 128, 134, 191] [10, 37, 50, 64, 118, 124, 129, 135] [11, 38, 51, 65, 119, 125, 130, 136] [39, 52, 66, 73, 120, 126, 131, 137] [40, 53, 67, 74, 121, 127, 132, 138] [1, 41, 54, 75, 122, 133, 139, 187] [3, 12, 42, 68, 101, 140, 147, 160] [13, 43, 56, 69, 141, 148, 154, 161] [14, 44, 57, 70, 142, 149, 155, 162] [15, 45, 58, 71, 143, 150, 156, 163] [46, 59, 72, 78, 144, 151, 157, 164] [2, 47, 60, 79, 145, 152, 158, 165] [3, 48, 61, 80, 99, 146, 153, 159] [5, 16, 49, 73, 134, 140, 147, 194] [17, 50, 62, 74, 128, 135, 141, 148] [18, 51, 63, 75, 129, 136, 142, 149] [19, 52, 64, 76, 130, 137, 143, 150] [53, 65, 77, 82, 131, 138, 144, 151] [4, 54, 66, 83, 132, 139, 145, 152] [5, 55, 67, 84, 133, 146, 153, 192] [8, 21, 56, 78, 106, 154, 160, 171] [22, 57, 68, 79, 155, 161, 166, 172] [23, 58, 69, 80, 156, 162, 167, 173] [24, 59, 70, 81, 157, 163, 168, 174] [6, 60, 71, 86, 158, 164, 169, 175] [7, 61, 72, 87, 102, 159, 165, 170] [11, 26, 62, 82, 140, 148, 154, 160] [27, 63, 73, 83, 141, 149, 155, 161] [28, 64, 74, 84, 142, 150, 156, 162] [29, 65, 75, 85, 143, 151, 157, 163] [9, 66, 76, 89, 144, 152, 158, 164] [10, 67, 77, 90, 145, 153, 159, 165] [15, 32, 68, 86, 113, 166, 171, 180] [33, 69, 78, 87, 167, 172, 176, 181] [34, 70, 79, 88, 168, 173, 177, 182] [12, 35, 71, 80, 169, 174, 178, 183] [13, 72, 81, 92, 107, 170, 175, 179] [19, 38, 73, 89, 154, 161, 166, 171] [39, 74, 82, 90, 155, 162, 167, 172] [40, 75, 83, 91, 156, 163, 168, 173] [16, 41, 76, 84, 157, 164, 169, 174] [17, 77, 85, 94, 158, 165, 170, 175] [24, 45, 78, 92, 122, 176, 180, 187] [46, 79, 86, 93, 177, 181, 184, 188] [20, 47, 80, 87, 178, 182, 185, 189] [21, 48, 81, 88, 114, 179, 183, 186] [29, 52, 82, 94, 166, 172, 176, 180] [53, 83, 89, 95, 167, 173, 177, 181] [25, 54, 84, 90, 168, 174, 178, 182] [26, 55, 85, 91, 169, 175, 179, 183] [35, 59, 86, 96, 133, 184, 187, 192] [30, 60, 87, 92, 185, 188, 190, 193] [31, 61, 88, 93, 123, 186, 189, 191] [41, 65, 89, 97, 176, 181, 184, 187] [36, 66, 90, 94, 177, 182, 185, 188] [37, 67, 91, 95, 178, 183, 186, 189] [42, 48, 71, 92, 146, 190, 192, 195] [43, 72, 93, 96, 134, 191, 193, 194] [49, 55, 76, 94, 184, 188, 190, 192] [50, 77, 95, 97, 185, 189, 191, 193] [0, 56, 81, 96, 98, 147, 194, 195] [1, 62, 85, 97, 190, 193, 194, 195]
Code ID 196-4-21 · download JSON · raw on GitHub