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[[196,12,17]] d ≤
n
196
k
12
d
17
kd²/n
17.694
w
7
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 17, d_Z ≤ 17 · 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 17 · witness weight 17 (claimed upper_bound)
witness operator (support, 17 qubits)
[10, 29, 34, 52, 55, 58, 69, 76, 89, 100, 115, 162, 165, 172, 180, 181, 187]
d_Z 17 · witness weight 17 (claimed upper_bound)
witness operator (support, 17 qubits)
[2, 6, 35, 38, 51, 58, 61, 81, 83, 86, 100, 106, 114, 156, 161, 179, 186]
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 (3,4): 98 (3,5): 931 (3,6): 9898 (3,7): 15239 (3,8): 5880 (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 (3,4): 98 (3,5): 931 (3,6): 9898 (3,7): 15239 (3,8): 5880 (3,9): 1764 (3,10): 392

Construction & provenance

authors @MathysRennela
provenance submitted through the challenge
novelty novelty not audited
construction Published 2BGA database row from arXiv:2306.16400v1: SmallGroup(98,3), a=1+g3+g14, b=1+g9+g10+g59.
builds on arXiv:2306.16400
date 2026-08-17
notes Published Lin–Pryadko baseline reconstructed from the authors public database at commit 403d194c3f98f0cadc236aecbc4a8b6139ccf23c. Paper reports exact d=17; repository records a witness-backed upper bound. Literature novelty is not claimed.
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

[[196,12,17]] — published 2BGA code on SmallGroup(98,3)

Direction & hypothesis

This is a literature reconstruction for the weight-8 × unrestricted cell. Lin--Pryadko's arXiv:2306.16400v1 database includes explicit non-abelian 2BGA rows up to group order 100; several were absent from the challenge board despite fitting its verification envelope.

What was searched

The authors' public database was audited at github.com/QEC-pages/2BGA-codes @ 403d194c3f98f0cadc236aecbc4a8b6139ccf23c. This row is SmallGroup(98,3) with database supports a=[3,14] and b=[9,10,59], using 1-based GAP positions excluding the identity. The identity was added and the group was rebuilt with GAP regular representations.

Evidence trail

The reconstructed matrices have n=196, k=12, maximum check weight 7 (therefore the repository's weight-8 class), and CSS commutation. The repository submission builder generated persisted X/Z logical witnesses of weight 17. verify/validate_candidate.py returned passed: true, no lighter logical in 8,000 RIS trials, no exact duplicate, no WL-equivalent duplicate, and board_advancing: true for weight-8 × unrestricted.

The paper reports d=17; this submission records only the witness-backed upper bound d <= 17 with upper_bound confidence. Literature novelty is not claimed: this is a published baseline reconstruction.

Dead ends

No search beyond reconstruction was performed. The row was selected because its parameter set was absent from the board comparison and it passed the computed Pareto check.

Tools

GAP for SmallGroup enumeration and the repository's research/kit/group_algebra.py, submit.py, and trusted validator. No changes to verify/ were made.

Reproduction

Use GAP SmallGroup(98,3) and add the identity to the database supports: a=1+g3+g14, b=1+g9+g10+g59, with the database's 1-based element positions. Construct H_X=[L(a)|R(b)] and H_Z=[R(b)^T|L(a)^T] over GF(2), then run the repository validator.

Parity checks

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