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[[196,6,20]] d ≤
n
196
k
6
d
20
kd²/n
12.245
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 20, d_Z ≤ 20 · w_X = 7, w_Z = 7 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[3, 7, 11, 45, 47, 53, 56, 75, 101, 102, 111, 131, 143, 147, 166, 167, 175, 183, 189, 193]
d_Z 20 · witness weight 20 (claimed upper_bound)
witness operator (support, 20 qubits)
[4, 9, 12, 20, 45, 52, 56, 66, 74, 76, 84, 89, 109, 110, 123, 145, 161, 164, 180, 193]
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 6 · H_Z 6 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 7 · H_Z 7
qubit degrees H_X 3–4 (mean 3.5) · H_Z 3–4 (mean 3.5)
trapping sets H_X (1,3)×98 (2,4)×294 (3,3)×98 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,3): 98 (1,4): 98 (2,4): 294 (2,5): 1176 (2,6): 588 (3,3): 98 (3,5): 1470 (3,6): 10388 (3,7): 13622 (3,8): 5292 (3,9): 1764 (3,10): 392
trapping sets H_Z (1,3)×98 (2,4)×294 (3,3)×98 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,3): 98 (1,4): 98 (2,4): 294 (2,5): 1176 (2,6): 588 (3,3): 98 (3,5): 1470 (3,6): 10388 (3,7): 13622 (3,8): 5292 (3,9): 1764 (3,10): 392

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction contributed via qldpc submit
model DeepSeek V4 Flash 0731 (claimed, not verified)
date 2026-09-18
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

[[196,6,20]] — two-block group-algebra code on SmallGroup(98,3)

Direction & hypothesis

Target: unrestricted weight-8 frontier for hackathon issue #1155. Reconstruct a published 2BGA code absent from the local board. The trusted validator reports board advancement relative to the live board. Efficiency kd²/n = 12.24.

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_wt7wtL3_order98_k6.txt, line 10. This row is SmallGroup(98,3) with nonidentity GAP supports a=[3, 32], b=[5, 16, 72]. 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=6), and no lighter logical was found in the refutation search. Distance is a witness-backed upper bound d ≤ 20, not an exact certificate. The published distance (d=20) 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

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