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[[168,16,15]] d ≤
n
168
k
16
d
15
kd²/n
21.429
w
8

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Distance

d_X 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[3, 6, 11, 14, 36, 50, 59, 81, 86, 88, 98, 103, 106, 127, 132]
d_Z 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[4, 19, 21, 28, 35, 56, 75, 92, 104, 112, 120, 124, 157, 161, 165]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors Aydin, Arda and Tamo, Itzhak and Barg, Alexander
provenance literature baseline
construction Coset-based two-block code Q_G^H(a,b) from arXiv:2606.17268, published data l=168, GAP SmallGroup id m=33, non-normal subgroup index s=1, a=[1,72,106,109], b=[1,11,18,26]. H_X=[A|B], H_Z=[B^T|A^T]. Reconstructed directly from the authors' released 84x84 permutation building blocks.
model classical construction (no AI model)
reference arXiv:2606.17268
date 2026-06-15
notes Published as exact d=15 in Table 2 and in the authors' data. Conservatively recorded here as a witness-backed upper bound pending this repository's own exact certification. Upstream data commit a828dc43c55982e0212febea634775d36bf6e968.
family 2BGA coset (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight ≤ 8 (computed)

How this code was found

no research note was staged with this submission — notes are requested for new submissions (notes/README.md)

Parity checks

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