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[[168,20,14]] d ≤
n
168
k
20
d
14
kd²/n
23.333
w
8

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Distance

d_X 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[13, 27, 40, 50, 51, 60, 61, 79, 80, 81, 137, 143, 149, 160]
d_Z 14 · witness weight 14 (claimed upper_bound)
witness operator (support, 14 qubits)
[5, 9, 13, 78, 99, 100, 109, 122, 137, 144, 145, 146, 164, 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 Regular two-block group-algebra code on GAP SmallGroup(84,3) = C7 x (C3 : C4) with a = Elements(G) indices [1,2,3,51], b = [1,5,36,60] (identity included, GAP Elements() order; Lin-Pryadko convention H_X=[A|B], A=sum L(a), B=sum R(b)). This is the double-cover lift (arXiv:2606.17268, Table 4) of the base [[84,16,8]] 2BGA on SmallGroup(42,3) = C7 x S3 (a=[1,2,3,29], b=[1,4,15,27]) through the central C2.
model classical construction (no AI model)
reference arXiv:2606.17268, arXiv:2306.16400
date 2026-06-15
notes Literature baseline, seeded 2026-07-01. Published in arXiv:2606.17268 (Table 4) as the double cover of the base [[84,16,8]] 2BGA; the paper states d=14 with no upper-bound qualifier (their exact-certified tier). Recorded here as a witness-backed upper bound (d<=14, both sides, found at 150k RIS trials) pending this repo's own server certification. Reproduction was validated end to end: the same GAP Elements() indexing and Lin-Pryadko 2BGA convention (arXiv:2306.16400) reproduce ten published Lin-Pryadko data rows (k and d) on SmallGroup(42,3) and SmallGroup(84,3), and ATB's base code [[84,16,8]] exactly. This baseline strictly dominates the n=180 coset submissions [[180,20,14]] and [[180,18,14]] at weight 8.
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, 6, 51, 54, 84, 96, 141, 160] [0, 1, 46, 60, 85, 101, 147, 163] [0, 2, 14, 63, 86, 95, 106, 152] [1, 3, 38, 66, 87, 107, 128, 151] [4, 16, 65, 67, 84, 88, 154, 166] [1, 2, 5, 58, 89, 100, 112, 157] [3, 6, 59, 71, 90, 113, 134, 156] [4, 7, 18, 72, 85, 91, 159, 167] [2, 8, 25, 73, 92, 105, 118, 160] [3, 5, 9, 50, 88, 93, 119, 140] [4, 10, 27, 75, 86, 94, 107, 162] [7, 11, 52, 76, 87, 95, 142, 161] [12, 29, 37, 77, 88, 96, 127, 150] [5, 8, 13, 69, 97, 111, 124, 163] [6, 9, 14, 70, 91, 98, 125, 146] [7, 10, 15, 30, 89, 99, 113, 165] [11, 16, 31, 79, 90, 100, 148, 164] [12, 17, 32, 80, 91, 101, 133, 155] [8, 18, 37, 79, 95, 102, 117, 130] [9, 13, 19, 62, 94, 103, 131, 151] [10, 20, 39, 81, 92, 104, 119, 166] [11, 15, 21, 64, 93, 96, 105, 153] [12, 22, 41, 49, 87, 94, 106, 139] [17, 23, 24, 82, 95, 107, 114, 138] [13, 18, 24, 77, 100, 108, 123, 136] [14, 19, 25, 78, 99, 109, 137, 156] [15, 20, 26, 42, 97, 110, 125, 167] [16, 21, 27, 43, 98, 101, 111, 158] [17, 22, 28, 44, 90, 99, 112, 145] [23, 29, 45, 83, 100, 113, 120, 144] [16, 18, 30, 49, 105, 114, 129, 142] [19, 24, 31, 72, 88, 104, 115, 143] [20, 32, 51, 83, 102, 107, 116, 131] [21, 26, 33, 74, 103, 106, 117, 161] [22, 34, 53, 61, 93, 104, 118, 150] [23, 28, 35, 36, 84, 105, 119, 126] [12, 24, 30, 36, 111, 120, 135, 148] [25, 31, 37, 82, 91, 110, 121, 149] [26, 32, 38, 54, 108, 113, 122, 137] [27, 33, 39, 55, 109, 112, 123, 164] [28, 34, 40, 56, 98, 110, 124, 155] [29, 35, 41, 57, 85, 111, 125, 132] [27, 30, 42, 61, 117, 126, 141, 153] [7, 31, 36, 43, 94, 116, 127, 154] [29, 32, 44, 63, 114, 119, 128, 143] [33, 38, 45, 80, 96, 115, 118, 129] [34, 46, 65, 71, 87, 103, 116, 130] [35, 40, 47, 48, 86, 117, 131, 138] [22, 36, 42, 48, 123, 132, 147, 158] [23, 37, 43, 49, 99, 122, 133, 159] [0, 38, 44, 50, 120, 125, 134, 149] [39, 45, 51, 66, 101, 121, 124, 135] [40, 46, 52, 67, 90, 109, 122, 136] [41, 47, 53, 68, 89, 123, 137, 144] [39, 42, 54, 71, 129, 138, 152, 161] [15, 43, 48, 55, 104, 128, 139, 162] [41, 44, 56, 73, 126, 131, 140, 154] [17, 45, 50, 57, 106, 127, 130, 141] [6, 46, 58, 75, 93, 115, 128, 142] [47, 52, 59, 60, 84, 92, 129, 143] [34, 48, 54, 60, 135, 144, 157, 164] [35, 49, 55, 61, 110, 134, 145, 165] [2, 50, 56, 62, 132, 137, 146, 159] [3, 51, 57, 63, 112, 133, 136, 147] [4, 52, 58, 64, 98, 121, 134, 148] [53, 59, 65, 76, 85, 97, 135, 149] [26, 55, 60, 66, 116, 140, 150, 166] [53, 56, 67, 79, 138, 143, 151, 162] [28, 57, 62, 68, 118, 139, 142, 152] [14, 58, 69, 81, 103, 127, 140, 153] [1, 59, 64, 70, 86, 102, 141, 154] [47, 61, 66, 71, 122, 146, 155, 167] [8, 62, 67, 72, 144, 149, 156, 165] [9, 63, 68, 73, 124, 145, 148, 157] [10, 64, 69, 74, 109, 133, 146, 158] [11, 65, 70, 75, 89, 108, 147, 159] [40, 68, 72, 76, 130, 150, 153, 160] [25, 69, 77, 83, 115, 139, 151, 161] [5, 70, 74, 78, 92, 114, 152, 162] [19, 73, 76, 79, 136, 155, 158, 163] [20, 74, 77, 80, 121, 145, 156, 164] [21, 75, 78, 81, 97, 120, 157, 165] [13, 78, 80, 82, 102, 126, 160, 166] [33, 81, 82, 83, 108, 132, 163, 167]
H_Z (84 checks, sparse supports)
[0, 4, 35, 59, 84, 85, 86, 134] [1, 7, 41, 65, 85, 87, 89, 154] [2, 10, 47, 70, 86, 89, 92, 146] [3, 11, 22, 46, 87, 90, 93, 147] [4, 9, 12, 31, 88, 91, 94, 148] [5, 15, 53, 75, 89, 93, 97, 162] [6, 16, 28, 52, 84, 90, 98, 142] [7, 14, 17, 37, 91, 95, 99, 127] [8, 20, 59, 78, 92, 97, 102, 156] [9, 21, 34, 58, 93, 98, 103, 157] [10, 19, 22, 43, 94, 99, 104, 158] [2, 11, 18, 23, 95, 100, 105, 159] [0, 12, 21, 45, 96, 101, 106, 120] [13, 26, 65, 81, 97, 103, 108, 166] [14, 27, 40, 64, 86, 98, 109, 153] [15, 25, 28, 49, 99, 105, 110, 139] [5, 16, 24, 29, 88, 100, 111, 114] [1, 17, 27, 51, 101, 107, 112, 141] [18, 32, 70, 82, 91, 102, 108, 114] [19, 33, 46, 69, 103, 109, 115, 163] [20, 31, 34, 55, 104, 110, 116, 164] [8, 21, 30, 35, 105, 111, 117, 165] [2, 22, 33, 57, 106, 112, 118, 132] [3, 10, 23, 32, 107, 113, 119, 133] [24, 38, 75, 83, 107, 108, 115, 120] [25, 39, 52, 74, 92, 109, 121, 161] [26, 37, 40, 61, 110, 117, 122, 150] [13, 27, 36, 41, 94, 111, 123, 126] [5, 28, 39, 63, 112, 119, 124, 152] [6, 15, 29, 38, 96, 113, 125, 128] [23, 30, 44, 78, 99, 114, 120, 126] [31, 45, 58, 77, 100, 115, 121, 127] [32, 43, 46, 66, 101, 116, 122, 128] [18, 33, 42, 47, 117, 123, 129, 167] [8, 34, 45, 68, 118, 124, 130, 144] [9, 20, 35, 44, 119, 125, 131, 145] [29, 36, 50, 81, 119, 120, 127, 132] [37, 51, 64, 80, 96, 102, 121, 133] [38, 49, 52, 71, 87, 122, 129, 134] [24, 39, 48, 53, 104, 123, 135, 138] [13, 40, 51, 73, 124, 131, 136, 160] [14, 26, 41, 50, 106, 125, 137, 140] [35, 42, 56, 82, 110, 126, 132, 138] [12, 43, 57, 69, 111, 127, 133, 139] [3, 44, 55, 58, 112, 128, 134, 140] [30, 45, 54, 59, 113, 129, 135, 141] [18, 46, 57, 76, 85, 130, 136, 142] [19, 32, 47, 56, 131, 137, 143, 155] [41, 48, 62, 83, 131, 132, 139, 144] [17, 49, 63, 74, 106, 114, 133, 145] [6, 50, 61, 64, 93, 134, 141, 146] [36, 51, 60, 65, 84, 116, 135, 147] [24, 52, 63, 79, 95, 136, 143, 148] [25, 38, 53, 62, 118, 137, 149, 151] [23, 47, 54, 67, 84, 122, 138, 144] [22, 55, 68, 77, 123, 139, 145, 150] [9, 56, 66, 69, 124, 140, 146, 151] [0, 42, 57, 70, 125, 141, 147, 152] [11, 30, 58, 68, 89, 142, 148, 153] [31, 44, 59, 67, 90, 143, 149, 154] [29, 53, 60, 72, 85, 143, 144, 150] [28, 61, 73, 80, 118, 126, 145, 155] [14, 62, 71, 74, 103, 146, 152, 156] [1, 48, 63, 75, 86, 128, 147, 157] [16, 36, 64, 73, 105, 148, 154, 158] [37, 50, 65, 72, 88, 130, 149, 159] [12, 34, 66, 76, 87, 135, 150, 155] [3, 19, 67, 77, 88, 136, 151, 156] [2, 54, 68, 78, 137, 152, 157, 160] [21, 42, 69, 76, 97, 153, 158, 161] [4, 43, 56, 70, 98, 154, 159, 162] [17, 40, 71, 79, 90, 130, 138, 155] [6, 25, 72, 80, 91, 115, 156, 160] [5, 60, 73, 81, 92, 140, 157, 163] [27, 48, 74, 79, 117, 158, 162, 164] [7, 49, 62, 75, 94, 142, 159, 165] [0, 8, 76, 82, 95, 149, 160, 163] [11, 33, 54, 77, 96, 108, 161, 164] [10, 55, 67, 78, 109, 162, 165, 166] [1, 13, 79, 83, 100, 102, 151, 163] [16, 39, 60, 80, 101, 129, 164, 166] [15, 61, 72, 81, 104, 153, 165, 167] [4, 20, 66, 82, 107, 121, 166, 167] [7, 26, 71, 83, 113, 116, 161, 167]
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