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[[196,2,22]] d ≤
n
196
k
2
d
22
kd²/n
4.939
w
8
X/Z
1

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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)
[2, 9, 19, 23, 30, 31, 41, 49, 50, 59, 75, 86, 101, 103, 121, 139, 145, 147, 164, 182, 185, 194]
d_Z 22 · witness weight 22 (claimed upper_bound)
witness operator (support, 22 qubits)
[6, 8, 18, 21, 24, 37, 61, 62, 67, 77, 98, 116, 120, 122, 138, 149, 155, 174, 175, 178, 179, 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)×245 (3,4)×392 (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): 245 (2,6): 2254 (3,4): 392 (3,6): 5684 (3,8): 39690 (3,10): 3136
trapping sets H_Z (1,4)×196 (2,4)×245 (3,4)×392 (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): 245 (2,6): 2254 (3,4): 392 (3,6): 5684 (3,8): 39690 (3,10): 3136

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_k2.txt, line 10. nonidentity supports a=[2, 3, 7], b=[9, 34, 80].
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

[[196,2,22]] — 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 = 4.94.

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_k2.txt, line 10. This row is SmallGroup(98,1) with nonidentity GAP supports a=[2, 3, 7], b=[9, 34, 80]. 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(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, 7], b=[9, 34, 80] 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) = 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 98 (max weight 8) · Z-checks 98 (max weight 8)
H_X (98 checks, sparse supports)
[0, 1, 93, 96, 98, 120, 133, 167] [0, 1, 4, 9, 99, 125, 139, 172] [0, 2, 4, 96, 100, 130, 145, 176] [3, 5, 30, 42, 101, 131, 146, 177] [2, 4, 9, 16, 102, 136, 152, 180] [3, 5, 10, 17, 103, 137, 153, 181] [0, 2, 6, 9, 104, 122, 142, 158] [3, 7, 10, 42, 105, 143, 159, 184] [8, 11, 43, 56, 98, 106, 144, 185] [6, 9, 16, 25, 107, 127, 149, 164] [7, 10, 17, 26, 108, 150, 165, 187] [8, 11, 18, 27, 99, 109, 151, 188] [2, 6, 12, 16, 110, 132, 155, 169] [3, 7, 13, 17, 111, 133, 156, 170] [8, 14, 18, 56, 100, 112, 157, 190] [15, 19, 57, 68, 101, 113, 158, 191] [12, 16, 25, 36, 114, 138, 161, 174] [13, 17, 26, 37, 115, 139, 162, 175] [14, 18, 27, 38, 102, 116, 163, 192] [15, 19, 28, 39, 103, 117, 164, 193] [6, 12, 20, 25, 118, 144, 166, 178] [7, 13, 21, 26, 119, 145, 167, 179] [8, 14, 22, 27, 104, 120, 146, 168] [15, 23, 28, 68, 105, 121, 169, 194] [24, 29, 69, 78, 106, 122, 128, 170] [20, 25, 36, 49, 123, 151, 171, 182] [21, 26, 37, 50, 124, 152, 172, 183] [22, 27, 38, 51, 107, 125, 153, 173] [23, 28, 39, 52, 108, 126, 174, 195] [24, 29, 40, 53, 109, 127, 134, 175] [12, 20, 30, 36, 113, 128, 157, 185] [13, 21, 31, 37, 129, 158, 176, 186] [14, 22, 32, 38, 110, 130, 159, 177] [15, 23, 33, 39, 98, 111, 131, 178] [24, 34, 40, 78, 112, 132, 140, 179] [35, 41, 79, 86, 104, 113, 133, 141] [5, 30, 36, 49, 117, 134, 163, 188] [31, 37, 50, 62, 135, 164, 180, 189] [32, 38, 51, 63, 114, 136, 165, 181] [33, 39, 52, 64, 99, 115, 137, 182] [34, 40, 53, 65, 116, 138, 147, 183] [35, 41, 54, 66, 107, 117, 139, 148] [20, 30, 42, 49, 121, 140, 168, 190] [21, 31, 43, 50, 122, 141, 169, 191] [22, 32, 44, 51, 118, 142, 170, 184] [23, 33, 45, 52, 100, 119, 143, 185] [24, 34, 46, 53, 101, 120, 144, 186] [35, 47, 54, 86, 110, 121, 145, 154] [48, 55, 87, 92, 111, 122, 146, 155] [5, 10, 42, 49, 126, 147, 173, 192] [11, 43, 50, 62, 127, 148, 174, 193] [44, 51, 63, 73, 123, 149, 175, 187] [45, 52, 64, 74, 102, 124, 150, 188] [46, 53, 65, 75, 103, 125, 151, 189] [47, 54, 66, 76, 114, 126, 152, 160] [48, 55, 67, 77, 115, 127, 153, 161] [31, 43, 56, 62, 132, 154, 178, 194] [32, 44, 57, 63, 128, 133, 155, 179] [33, 45, 58, 64, 104, 129, 156, 190] [34, 46, 59, 65, 105, 130, 157, 191] [35, 47, 60, 66, 106, 118, 131, 158] [48, 61, 67, 92, 119, 132, 159, 166] [11, 18, 56, 62, 138, 160, 182, 195] [19, 57, 63, 73, 134, 139, 161, 183] [58, 64, 74, 82, 107, 135, 162, 192] [59, 65, 75, 83, 108, 136, 163, 193] [60, 66, 76, 84, 109, 123, 137, 164] [61, 67, 77, 85, 124, 138, 165, 171] [44, 57, 68, 73, 140, 145, 166, 186] [45, 58, 69, 74, 110, 141, 146, 167] [46, 59, 70, 75, 111, 142, 168, 194] [47, 60, 71, 76, 112, 128, 143, 169] [48, 61, 72, 77, 113, 129, 144, 170] [19, 28, 68, 73, 147, 152, 171, 189] [29, 69, 74, 82, 114, 148, 153, 172] [70, 75, 83, 89, 115, 149, 173, 195] [71, 76, 84, 90, 116, 134, 150, 174] [72, 77, 85, 91, 117, 135, 151, 175] [58, 69, 78, 82, 118, 154, 159, 176] [59, 70, 79, 83, 98, 119, 155, 177] [60, 71, 80, 84, 120, 140, 156, 178] [61, 72, 81, 85, 121, 141, 157, 179] [29, 40, 78, 82, 123, 160, 165, 180] [41, 79, 83, 89, 99, 124, 161, 181] [80, 84, 90, 94, 125, 147, 162, 182] [81, 85, 91, 95, 126, 148, 163, 183] [70, 79, 86, 89, 100, 129, 166, 184] [71, 80, 87, 90, 101, 130, 167, 185] [72, 81, 88, 91, 131, 154, 168, 186] [41, 54, 86, 89, 102, 135, 171, 187] [55, 87, 90, 94, 103, 136, 172, 188] [88, 91, 95, 97, 137, 160, 173, 189] [80, 87, 92, 94, 105, 142, 176, 190] [81, 88, 93, 95, 106, 143, 177, 191] [55, 67, 92, 94, 108, 149, 180, 192] [1, 93, 95, 97, 109, 150, 181, 193] [88, 93, 96, 97, 112, 156, 184, 194] [1, 4, 96, 97, 116, 162, 187, 195]
H_Z (98 checks, sparse supports)
[0, 8, 33, 79, 98, 99, 100, 104] [1, 11, 39, 83, 98, 99, 193, 195] [2, 14, 45, 86, 100, 102, 104, 110] [3, 15, 46, 87, 101, 103, 105, 111] [4, 18, 52, 89, 99, 100, 102, 195] [5, 19, 53, 90, 101, 103, 134, 147] [6, 22, 35, 58, 104, 107, 110, 118] [7, 23, 59, 92, 105, 108, 111, 119] [8, 24, 60, 93, 106, 109, 112, 120] [9, 27, 41, 64, 99, 102, 104, 107] [10, 28, 65, 94, 103, 105, 108, 147] [11, 29, 66, 95, 106, 109, 148, 160] [12, 32, 47, 69, 110, 114, 118, 128] [13, 33, 48, 70, 111, 115, 119, 129] [14, 34, 71, 96, 112, 116, 120, 130] [15, 30, 35, 72, 113, 117, 121, 131] [16, 38, 54, 74, 102, 107, 110, 114] [17, 39, 55, 75, 103, 108, 111, 115] [18, 40, 76, 97, 109, 112, 116, 160] [19, 36, 41, 77, 113, 117, 161, 171] [20, 44, 60, 78, 118, 123, 128, 140] [21, 45, 61, 79, 119, 124, 129, 141] [0, 22, 46, 80, 120, 125, 130, 142] [23, 42, 47, 81, 121, 126, 131, 143] [6, 24, 43, 48, 122, 127, 132, 144] [25, 51, 66, 82, 107, 114, 118, 123] [26, 52, 67, 83, 108, 115, 119, 124] [1, 27, 53, 84, 109, 116, 120, 125] [28, 49, 54, 85, 117, 121, 126, 171] [9, 29, 50, 55, 122, 127, 172, 180] [24, 30, 57, 71, 101, 128, 134, 140] [31, 58, 72, 86, 129, 135, 141, 154] [2, 32, 59, 87, 130, 136, 142, 155] [3, 33, 60, 88, 131, 137, 143, 156] [12, 34, 56, 61, 132, 138, 144, 157] [0, 13, 35, 57, 133, 139, 145, 158] [29, 36, 63, 76, 114, 123, 128, 134] [37, 64, 77, 89, 115, 124, 129, 135] [4, 38, 65, 90, 116, 125, 130, 136] [5, 39, 66, 91, 117, 126, 131, 137] [16, 40, 62, 67, 127, 132, 138, 180] [1, 17, 41, 63, 133, 139, 181, 187] [34, 42, 68, 80, 101, 105, 140, 147] [35, 43, 69, 81, 106, 141, 148, 154] [6, 44, 70, 92, 142, 149, 155, 166] [7, 45, 71, 93, 143, 150, 156, 167] [8, 20, 46, 72, 144, 151, 157, 168] [2, 21, 47, 68, 145, 152, 158, 169] [3, 22, 48, 69, 146, 153, 159, 170] [40, 49, 73, 84, 123, 134, 140, 147] [41, 50, 74, 85, 124, 135, 141, 148] [9, 51, 75, 94, 125, 136, 142, 149] [10, 52, 76, 95, 126, 137, 143, 150] [11, 25, 53, 77, 127, 138, 144, 151] [4, 26, 54, 73, 139, 145, 152, 187] [5, 27, 55, 74, 146, 153, 188, 192] [47, 56, 78, 88, 106, 112, 154, 160] [12, 48, 57, 79, 113, 155, 161, 166] [13, 58, 80, 96, 156, 162, 167, 176] [14, 30, 59, 81, 157, 163, 168, 177] [6, 15, 31, 60, 158, 164, 169, 178] [7, 32, 61, 78, 159, 165, 170, 179] [54, 62, 82, 91, 135, 148, 154, 160] [16, 55, 63, 83, 136, 149, 155, 161] [17, 64, 84, 97, 137, 150, 156, 162] [18, 36, 65, 85, 138, 151, 157, 163] [9, 19, 37, 66, 139, 152, 158, 164] [10, 38, 67, 82, 153, 159, 165, 192] [20, 61, 68, 86, 113, 121, 166, 171] [0, 21, 69, 87, 122, 167, 172, 176] [22, 42, 70, 88, 168, 173, 177, 184] [12, 23, 43, 71, 169, 174, 178, 185] [13, 24, 44, 72, 170, 175, 179, 186] [25, 67, 73, 89, 149, 161, 166, 171] [1, 26, 74, 90, 150, 162, 167, 172] [27, 49, 75, 91, 151, 163, 168, 173] [16, 28, 50, 76, 152, 164, 169, 174] [17, 29, 51, 77, 153, 165, 170, 175] [2, 31, 78, 92, 122, 132, 176, 180] [3, 32, 79, 93, 133, 177, 181, 184] [20, 33, 56, 80, 178, 182, 185, 190] [21, 34, 57, 81, 179, 183, 186, 191] [4, 37, 82, 94, 162, 172, 176, 180] [5, 38, 83, 95, 163, 173, 177, 181] [25, 39, 62, 84, 164, 174, 178, 182] [26, 40, 63, 85, 165, 175, 179, 183] [7, 44, 86, 96, 133, 145, 184, 187] [8, 30, 45, 87, 146, 185, 188, 190] [31, 46, 68, 88, 186, 189, 191, 194] [10, 51, 89, 97, 173, 181, 184, 187] [11, 36, 52, 90, 174, 182, 185, 188] [37, 53, 73, 91, 175, 183, 186, 189] [14, 42, 58, 92, 146, 159, 190, 192] [15, 43, 59, 93, 98, 191, 193, 194] [18, 49, 64, 94, 182, 188, 190, 192] [19, 50, 65, 95, 183, 189, 191, 193] [23, 56, 70, 96, 98, 100, 194, 195] [28, 62, 75, 97, 189, 193, 194, 195]
Code ID 196-2-22 · download JSON · raw on GitHub