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[[144,8,16]] d ≤
n
144
k
8
d
16
kd²/n
14.222
w
10

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Distance

d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[4, 41, 43, 54, 57, 68, 69, 79, 82, 97, 101, 109, 116, 117, 127, 140]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[2, 4, 13, 15, 27, 28, 29, 30, 46, 50, 63, 72, 104, 105, 121, 140]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Construction & provenance

authors @mathysrennela
provenance submitted through the challenge
novelty novelty not audited
construction 2BGA (arXiv:2306.16400) on GAP C9xD8, weight-5 random supports a=[7, 19, 36, 59, 69] b=[7, 41, 45, 48, 56]
model Xiaomi MiMo-V2.5 (claimed, not verified)
builds on arXiv:2306.16400
date 2026-07-29
notes Found by GAP-enhanced autoresearch (see fieldnotes/2026-07-28-gap-campaign-results.md). 2BGA code on C9xD8, weight-5 random supports. Distance upper bound confirmed flat across 400->1M RIS trials. Literature novelty unverified.
family generalized bicycle (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 72 · Z-checks 72
H_X (72 checks, sparse supports)
[6, 22, 43, 60, 68, 96, 127, 132, 133, 134] [11, 14, 33, 49, 52, 88, 119, 124, 125, 138] [1, 37, 56, 61, 67, 92, 123, 129, 131, 143] [1, 18, 41, 59, 71, 73, 112, 115, 122, 141] [7, 19, 23, 57, 69, 81, 116, 119, 138, 141] [20, 21, 37, 43, 61, 93, 94, 117, 139, 142] [4, 27, 34, 48, 63, 84, 121, 123, 131, 142] [0, 27, 31, 51, 63, 72, 102, 105, 130, 137] [2, 3, 15, 64, 65, 77, 106, 127, 134, 137] [10, 13, 29, 33, 53, 85, 86, 107, 135, 143] [6, 7, 54, 66, 69, 91, 107, 110, 117, 139] [8, 18, 38, 44, 71, 104, 111, 114, 139, 140] [9, 22, 36, 56, 68, 90, 112, 126, 137, 141] [7, 19, 24, 36, 45, 79, 80, 129, 131, 136] [8, 21, 37, 42, 67, 80, 95, 102, 130, 132] [9, 38, 54, 59, 69, 78, 81, 83, 113, 133] [18, 23, 39, 41, 44, 79, 97, 113, 135, 143] [22, 40, 42, 56, 61, 80, 108, 114, 115, 118] [2, 14, 46, 52, 62, 83, 100, 107, 117, 135] [0, 10, 28, 53, 70, 101, 104, 114, 135, 136] [3, 16, 26, 48, 64, 82, 102, 118, 137, 141] [2, 3, 16, 35, 46, 75, 76, 121, 123, 140] [4, 10, 13, 32, 70, 76, 87, 112, 122, 124] [5, 26, 48, 51, 65, 74, 77, 91, 103, 125] [12, 15, 27, 31, 34, 75, 89, 103, 139, 142] [14, 28, 32, 50, 53, 76, 98, 104, 105, 126] [8, 18, 21, 39, 58, 78, 109, 125, 127, 133] [19, 22, 36, 55, 60, 78, 89, 97, 128, 143] [20, 23, 41, 66, 71, 76, 80, 108, 111, 129] [7, 24, 38, 54, 57, 108, 109, 124, 130, 132] [25, 37, 43, 55, 68, 78, 103, 110, 113, 138] [6, 9, 22, 25, 54, 73, 94, 114, 121, 140] [8, 23, 39, 45, 55, 90, 95, 96, 99, 141] [24, 36, 42, 56, 60, 74, 91, 93, 96, 116] [25, 41, 57, 66, 69, 92, 94, 97, 101, 131] [7, 38, 44, 58, 59, 73, 95, 98, 126, 131] [0, 10, 29, 31, 50, 74, 99, 119, 125, 133] [3, 11, 26, 47, 65, 74, 89, 97, 120, 142] [12, 13, 33, 62, 70, 76, 80, 98, 101, 121] [5, 14, 28, 46, 49, 98, 99, 122, 124, 132] [15, 27, 35, 47, 64, 74, 100, 103, 113, 134] [5, 11, 14, 17, 26, 72, 86, 104, 129, 136] [4, 12, 15, 35, 62, 82, 87, 88, 109, 137] [16, 28, 32, 46, 52, 78, 83, 85, 88, 106] [17, 31, 47, 64, 65, 84, 86, 89, 111, 123] [3, 30, 34, 48, 51, 72, 87, 108, 118, 123] [1, 20, 37, 40, 41, 90, 91, 106, 116, 138] [19, 24, 38, 42, 58, 86, 92, 94, 128, 139] [21, 39, 43, 55, 67, 72, 73, 92, 126, 140] [40, 44, 54, 59, 71, 88, 90, 96, 127, 130] [18, 23, 45, 57, 66, 85, 91, 93, 127, 128] [1, 20, 25, 43, 66, 81, 82, 109, 115, 132] [8, 24, 44, 58, 67, 110, 113, 116, 117, 120] [21, 45, 55, 60, 68, 79, 84, 111, 114, 117] [4, 12, 13, 27, 30, 82, 83, 106, 116, 134] [5, 11, 28, 50, 51, 84, 86, 94, 120, 135] [15, 29, 31, 47, 63, 72, 73, 84, 118, 136] [30, 32, 46, 53, 70, 82, 88, 96, 119, 122] [10, 17, 33, 49, 62, 83, 85, 93, 119, 120] [0, 17, 29, 33, 47, 77, 90, 99, 105, 124] [4, 16, 30, 34, 70, 100, 103, 106, 107, 128] [13, 35, 49, 52, 62, 75, 92, 101, 104, 107] [6, 9, 40, 56, 59, 75, 79, 110, 111, 143] [19, 39, 45, 57, 60, 105, 108, 112, 115, 140] [36, 42, 58, 61, 67, 77, 81, 109, 112, 138] [1, 9, 40, 61, 71, 93, 97, 100, 128, 133] [2, 16, 30, 32, 48, 75, 79, 100, 101, 142] [11, 17, 29, 49, 65, 98, 102, 105, 115, 136] [26, 34, 50, 51, 63, 77, 81, 99, 102, 134] [0, 5, 50, 53, 63, 85, 89, 110, 120, 125] [6, 20, 25, 68, 69, 87, 95, 126, 129, 130] [2, 12, 35, 52, 64, 87, 95, 118, 121, 122]
H_Z (72 checks, sparse supports)
[7, 41, 45, 48, 56, 79, 91, 108, 131, 141] [3, 31, 35, 48, 56, 74, 75, 118, 123, 137] [23, 33, 36, 37, 40, 80, 90, 93, 138, 143] [21, 24, 61, 62, 66, 80, 92, 93, 109, 117] [21, 22, 25, 28, 38, 78, 94, 114, 126, 132] [8, 23, 59, 64, 68, 95, 111, 113, 127, 141] [15, 26, 27, 30, 43, 72, 82, 103, 134, 142] [13, 16, 53, 62, 66, 76, 82, 85, 101, 107] [13, 14, 17, 28, 38, 83, 86, 98, 104, 124] [4, 15, 51, 64, 68, 84, 87, 103, 134, 137] [20, 42, 51, 54, 57, 81, 91, 94, 108, 130] [15, 18, 43, 54, 58, 73, 109, 113, 127, 139] [6, 44, 53, 55, 56, 96, 110, 114, 126, 143] [9, 43, 50, 58, 69, 81, 94, 110, 126, 133] [9, 41, 44, 47, 55, 73, 90, 97, 111, 113] [22, 42, 45, 70, 71, 80, 96, 112, 114, 128] [1, 42, 43, 49, 57, 92, 93, 115, 132, 138] [24, 27, 37, 44, 69, 113, 116, 130, 131, 139] [12, 32, 46, 49, 59, 75, 83, 88, 98, 122] [10, 23, 33, 46, 50, 76, 85, 99, 119, 135] [2, 34, 47, 48, 61, 77, 100, 118, 123, 142] [5, 33, 50, 58, 65, 77, 86, 98, 120, 125] [5, 31, 34, 47, 55, 72, 84, 89, 99, 103] [14, 32, 35, 70, 71, 76, 88, 100, 104, 122] [0, 32, 33, 49, 57, 85, 101, 105, 119, 124] [16, 27, 34, 37, 65, 102, 103, 106, 123, 142] [25, 35, 38, 39, 67, 92, 95, 109, 113, 140] [32, 36, 39, 59, 68, 78, 79, 96, 112, 126] [18, 40, 60, 65, 66, 91, 97, 111, 115, 127] [19, 34, 38, 61, 66, 81, 108, 128, 131, 139] [7, 14, 20, 67, 68, 117, 126, 129, 132, 138] [23, 24, 30, 40, 60, 79, 96, 108, 116, 128] [11, 19, 25, 41, 61, 94, 97, 115, 129, 138] [7, 25, 59, 63, 67, 73, 81, 110, 130, 131] [8, 43, 46, 54, 60, 78, 96, 117, 132, 140] [9, 10, 18, 60, 61, 93, 112, 114, 133, 143] [17, 28, 29, 45, 63, 84, 85, 99, 105, 136] [26, 29, 42, 51, 64, 74, 77, 86, 102, 118] [10, 30, 52, 62, 69, 83, 87, 101, 107, 119] [11, 28, 44, 53, 62, 88, 98, 104, 120, 135] [3, 12, 22, 63, 64, 89, 118, 121, 134, 137] [15, 16, 30, 40, 52, 75, 88, 100, 106, 118] [11, 17, 19, 31, 53, 86, 89, 105, 119, 136] [3, 17, 51, 63, 67, 72, 77, 102, 120, 123] [4, 33, 46, 52, 54, 83, 88, 107, 121, 124] [5, 10, 18, 52, 53, 85, 104, 122, 125, 135] [17, 20, 45, 56, 71, 90, 93, 111, 115, 129] [1, 4, 36, 57, 58, 109, 112, 116, 128, 131] [37, 52, 55, 58, 69, 78, 92, 95, 117, 138] [6, 21, 31, 38, 71, 73, 111, 130, 133, 139] [3, 22, 39, 57, 71, 97, 108, 127, 140, 141] [2, 6, 21, 44, 45, 79, 95, 117, 127, 140] [1, 22, 29, 39, 59, 73, 90, 115, 133, 143] [1, 23, 26, 36, 69, 81, 91, 97, 129, 141] [12, 25, 35, 48, 70, 82, 87, 101, 103, 121] [0, 8, 26, 49, 50, 99, 102, 104, 120, 125] [27, 47, 50, 60, 65, 74, 84, 89, 105, 134] [2, 13, 28, 41, 70, 76, 101, 106, 122, 135] [7, 14, 29, 49, 70, 98, 107, 119, 124, 136] [2, 6, 13, 34, 35, 75, 87, 107, 121, 134] [0, 14, 29, 39, 51, 72, 99, 105, 125, 135] [0, 15, 26, 36, 65, 74, 77, 89, 136, 137] [0, 8, 40, 54, 68, 90, 110, 114, 130, 133] [9, 16, 18, 19, 55, 78, 79, 128, 140, 141] [13, 19, 41, 56, 67, 80, 92, 112, 116, 143] [7, 8, 12, 20, 42, 80, 95, 109, 116, 139] [1, 4, 30, 46, 64, 82, 100, 106, 122, 123] [5, 10, 11, 24, 47, 74, 86, 120, 124, 136] [11, 21, 31, 48, 63, 72, 84, 102, 125, 142] [3, 4, 12, 20, 32, 76, 82, 87, 106, 142] [5, 6, 24, 37, 66, 91, 94, 110, 129, 132] [2, 9, 16, 27, 62, 75, 83, 100, 121, 137]
raw submission JSON