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[[140,8,16]] d ≤
n
140
k
8
d
16
kd²/n
14.629
w
12

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Distance

d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[1, 8, 10, 12, 23, 28, 30, 32, 33, 75, 83, 116, 120, 124, 125, 131]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[2, 6, 7, 10, 13, 15, 46, 81, 94, 95, 96, 98, 100, 105, 117, 120]
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 C70, weight-6 random supports a=[36, 44, 47, 52, 60, 63] b=[8, 23, 35, 44, 63, 67]
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 C70, weight-6 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 70 · Z-checks 70
H_X (70 checks, sparse supports)
[3, 21, 26, 31, 46, 56, 73, 87, 96, 104, 127, 128] [5, 21, 26, 36, 41, 52, 75, 83, 91, 109, 123, 131] [7, 30, 35, 40, 54, 62, 77, 94, 96, 105, 113, 133] [8, 31, 36, 41, 55, 63, 74, 78, 97, 106, 114, 134] [10, 30, 35, 45, 50, 59, 80, 91, 99, 100, 118, 130] [11, 31, 36, 46, 51, 60, 72, 81, 92, 101, 119, 136] [3, 5, 8, 13, 19, 66, 75, 83, 103, 105, 122, 137] [14, 40, 45, 50, 61, 67, 79, 84, 104, 106, 115, 123] [9, 15, 41, 46, 51, 62, 70, 80, 85, 107, 116, 138] [3, 5, 11, 15, 17, 64, 73, 87, 100, 108, 126, 135] [18, 40, 45, 54, 58, 65, 76, 88, 101, 109, 110, 127] [6, 19, 41, 46, 55, 59, 71, 77, 89, 102, 111, 139] [7, 10, 14, 21, 28, 39, 75, 80, 91, 112, 129, 139] [8, 11, 15, 22, 29, 69, 81, 86, 92, 113, 115, 130] [16, 23, 50, 54, 58, 66, 72, 87, 93, 114, 116, 124] [17, 24, 51, 55, 59, 67, 73, 88, 90, 94, 117, 125] [7, 10, 18, 23, 26, 34, 73, 77, 96, 117, 132, 138] [8, 11, 19, 24, 27, 68, 78, 82, 97, 110, 118, 133] [12, 28, 50, 54, 61, 64, 74, 83, 98, 111, 119, 120] [13, 29, 51, 55, 62, 65, 75, 84, 95, 99, 112, 121] [13, 17, 22, 30, 37, 48, 80, 87, 100, 119, 121, 134] [14, 18, 23, 31, 38, 49, 81, 88, 95, 101, 122, 135] [15, 19, 24, 25, 32, 39, 76, 89, 96, 102, 123, 124] [26, 33, 58, 61, 64, 69, 70, 77, 97, 103, 125, 131] [16, 27, 34, 59, 62, 65, 78, 98, 100, 104, 126, 132] [13, 17, 27, 32, 35, 43, 77, 83, 105, 114, 125, 136] [14, 18, 28, 33, 36, 44, 78, 84, 90, 106, 126, 137] [15, 19, 20, 29, 34, 37, 79, 85, 91, 107, 120, 127] [21, 38, 58, 61, 66, 68, 71, 80, 92, 108, 121, 128] [12, 22, 39, 59, 62, 67, 81, 93, 105, 109, 122, 129] [22, 27, 32, 40, 47, 57, 71, 88, 97, 110, 129, 138] [1, 23, 28, 33, 41, 48, 82, 89, 98, 105, 111, 130] [24, 29, 34, 35, 42, 49, 72, 83, 99, 106, 112, 131] [25, 36, 43, 64, 66, 68, 73, 84, 107, 113, 132, 136] [12, 26, 37, 44, 65, 67, 85, 108, 110, 114, 133, 137] [22, 27, 37, 42, 45, 53, 70, 84, 92, 115, 132, 139] [0, 23, 28, 38, 43, 46, 85, 86, 93, 100, 116, 133] [24, 29, 30, 39, 44, 47, 74, 87, 94, 101, 117, 128] [20, 31, 48, 64, 66, 69, 75, 88, 102, 118, 129, 134] [16, 21, 32, 49, 65, 67, 89, 103, 115, 119, 130, 135] [4, 32, 37, 42, 50, 56, 75, 90, 98, 107, 120, 135] [5, 33, 38, 43, 51, 57, 76, 91, 99, 108, 115, 121] [1, 34, 39, 44, 45, 52, 77, 92, 109, 116, 122, 136] [20, 35, 46, 53, 68, 69, 78, 93, 117, 123, 137, 139] [0, 12, 16, 21, 36, 47, 70, 79, 86, 94, 118, 120] [2, 32, 37, 47, 52, 54, 73, 93, 95, 102, 124, 137] [3, 33, 38, 48, 53, 55, 79, 94, 96, 103, 110, 125] [0, 34, 39, 40, 49, 56, 80, 97, 104, 111, 126, 134] [25, 30, 41, 57, 68, 69, 81, 98, 112, 127, 135, 138] [1, 12, 16, 26, 31, 42, 71, 76, 82, 99, 113, 124] [10, 42, 47, 52, 58, 63, 81, 82, 100, 108, 117, 128] [4, 11, 43, 48, 53, 59, 83, 101, 109, 118, 124, 129] [0, 5, 44, 49, 54, 60, 84, 102, 119, 125, 130, 139] [2, 20, 25, 30, 45, 55, 72, 85, 86, 95, 103, 126] [7, 42, 47, 56, 60, 61, 78, 86, 103, 105, 112, 131] [2, 8, 43, 48, 57, 62, 87, 104, 106, 113, 120, 132] [1, 3, 44, 49, 50, 63, 88, 107, 114, 121, 133, 138] [4, 20, 25, 35, 40, 51, 74, 82, 89, 90, 108, 122] [9, 18, 52, 56, 60, 64, 89, 91, 110, 118, 126, 134] [2, 10, 19, 53, 57, 65, 92, 111, 119, 127, 131, 135] [0, 1, 3, 6, 11, 61, 71, 76, 93, 95, 112, 132] [6, 14, 52, 56, 63, 66, 85, 96, 113, 115, 122, 136] [4, 7, 15, 53, 57, 67, 97, 114, 116, 123, 128, 137] [0, 1, 5, 8, 9, 58, 70, 79, 90, 98, 117, 129] [6, 17, 28, 60, 63, 68, 99, 101, 120, 127, 133, 138] [2, 4, 7, 12, 18, 29, 71, 74, 82, 102, 121, 136] [9, 13, 23, 60, 63, 69, 94, 106, 123, 124, 130, 139] [2, 4, 10, 14, 16, 24, 70, 72, 86, 107, 125, 134] [6, 9, 13, 20, 27, 38, 74, 79, 90, 109, 111, 128] [6, 9, 17, 22, 25, 33, 72, 76, 95, 104, 116, 131]
H_Z (70 checks, sparse supports)
[8, 23, 35, 44, 63, 67, 106, 114, 117, 122, 130, 133] [11, 28, 30, 49, 60, 65, 101, 112, 119, 126, 130, 133] [5, 14, 32, 53, 67, 69, 115, 123, 125, 129, 135, 137] [0, 9, 15, 16, 33, 45, 70, 76, 79, 116, 126, 130] [3, 18, 37, 57, 65, 68, 110, 121, 127, 132, 135, 137] [1, 6, 12, 19, 38, 40, 71, 76, 79, 111, 122, 133] [10, 22, 41, 49, 60, 69, 81, 130, 131, 134, 138, 139] [2, 11, 16, 23, 25, 42, 72, 82, 86, 124, 132, 135] [3, 17, 24, 26, 43, 54, 73, 76, 83, 87, 125, 133] [7, 27, 44, 46, 63, 68, 78, 128, 133, 136, 138, 139] [4, 8, 12, 20, 28, 47, 74, 82, 86, 120, 129, 137] [5, 13, 21, 29, 48, 50, 75, 79, 83, 87, 121, 130] [17, 31, 49, 50, 57, 65, 88, 99, 104, 114, 119, 135] [1, 6, 18, 25, 32, 51, 76, 89, 90, 95, 136, 138] [7, 19, 26, 33, 35, 52, 77, 82, 91, 96, 131, 137] [8, 27, 34, 36, 53, 61, 78, 79, 83, 92, 97, 132] [13, 36, 44, 53, 54, 67, 84, 94, 109, 114, 119, 137] [0, 9, 14, 20, 37, 55, 79, 85, 90, 95, 134, 139] [10, 15, 21, 30, 38, 56, 80, 86, 91, 96, 128, 135] [11, 22, 31, 39, 57, 58, 76, 81, 87, 92, 97, 129] [15, 26, 40, 57, 63, 68, 97, 108, 113, 123, 127, 138] [1, 4, 12, 27, 41, 58, 70, 71, 82, 98, 109, 114] [5, 13, 28, 35, 42, 59, 83, 90, 99, 100, 105, 139] [14, 29, 36, 43, 45, 60, 84, 86, 91, 101, 106, 136] [2, 15, 37, 44, 46, 66, 85, 87, 92, 102, 107, 137] [19, 21, 45, 53, 60, 69, 92, 103, 118, 123, 127, 139] [0, 2, 16, 22, 46, 61, 70, 71, 86, 93, 104, 119] [3, 17, 23, 30, 47, 62, 87, 94, 95, 100, 105, 138] [18, 24, 31, 40, 48, 63, 82, 88, 96, 101, 106, 134] [4, 19, 32, 41, 49, 64, 83, 89, 97, 102, 107, 135] [4, 9, 20, 24, 36, 50, 72, 74, 90, 107, 118, 123] [5, 10, 21, 37, 51, 64, 70, 73, 75, 91, 108, 119] [11, 22, 38, 45, 52, 65, 92, 95, 100, 109, 110, 115] [6, 23, 39, 46, 53, 54, 93, 96, 101, 111, 116, 139] [0, 7, 24, 47, 55, 69, 86, 94, 97, 102, 112, 117] [2, 6, 25, 29, 31, 54, 72, 74, 95, 102, 113, 127] [3, 7, 26, 32, 55, 66, 71, 73, 75, 96, 103, 114] [8, 27, 33, 40, 56, 67, 90, 97, 104, 105, 110, 115] [9, 28, 34, 41, 50, 57, 91, 98, 106, 111, 116, 138] [1, 10, 29, 42, 51, 68, 82, 92, 99, 107, 112, 117] [10, 17, 30, 34, 46, 58, 72, 77, 80, 100, 117, 127] [11, 18, 31, 47, 59, 68, 71, 73, 78, 81, 101, 118] [12, 19, 32, 48, 54, 60, 102, 105, 110, 119, 120, 124] [2, 13, 33, 49, 55, 61, 95, 103, 106, 111, 121, 125] [3, 14, 25, 34, 56, 62, 96, 104, 107, 112, 122, 126] [7, 13, 35, 39, 41, 61, 74, 77, 80, 105, 112, 123] [8, 14, 36, 42, 62, 69, 70, 75, 78, 81, 106, 113] [15, 16, 37, 43, 50, 63, 100, 107, 114, 115, 120, 124] [4, 17, 38, 44, 51, 58, 90, 101, 108, 116, 121, 125] [5, 18, 20, 39, 52, 59, 91, 102, 109, 117, 122, 126] [18, 27, 40, 44, 55, 64, 74, 77, 84, 88, 110, 126] [19, 20, 28, 41, 56, 65, 75, 78, 85, 89, 111, 127] [6, 21, 29, 42, 57, 61, 71, 112, 115, 120, 128, 131] [1, 7, 22, 43, 62, 66, 105, 113, 116, 121, 129, 132] [14, 22, 45, 49, 51, 66, 72, 80, 84, 88, 115, 122] [15, 23, 25, 46, 52, 67, 73, 81, 85, 89, 116, 123] [9, 24, 26, 47, 53, 58, 70, 110, 117, 124, 128, 131] [0, 10, 27, 48, 59, 64, 100, 111, 118, 125, 129, 132] [0, 28, 37, 50, 62, 68, 80, 84, 93, 98, 120, 133] [12, 29, 30, 38, 51, 63, 74, 81, 85, 94, 99, 121] [4, 13, 31, 39, 52, 66, 75, 122, 124, 128, 134, 136] [1, 23, 32, 54, 59, 69, 77, 88, 93, 98, 124, 130] [16, 24, 33, 35, 55, 60, 72, 78, 89, 94, 99, 125] [2, 17, 34, 36, 56, 64, 73, 120, 126, 131, 134, 136] [3, 20, 38, 47, 58, 67, 79, 88, 93, 103, 108, 128] [9, 21, 39, 40, 48, 59, 80, 89, 94, 104, 109, 129] [5, 25, 33, 42, 61, 65, 76, 84, 98, 103, 108, 131] [6, 26, 34, 43, 45, 62, 77, 85, 99, 104, 109, 132] [8, 16, 30, 48, 56, 64, 87, 98, 103, 113, 118, 134] [11, 12, 35, 43, 52, 66, 83, 93, 108, 113, 118, 136]
raw submission JSON