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[[136,6,17]] d ≤
n
136
k
6
d
17
kd²/n
12.75
w
12

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Distance

d_X 17 · witness weight 17 (claimed upper_bound)
witness operator (support, 17 qubits)
[24, 37, 38, 39, 40, 41, 65, 67, 72, 90, 97, 99, 100, 103, 119, 125, 126]
d_Z 17 · witness weight 17 (claimed upper_bound)
witness operator (support, 17 qubits)
[5, 10, 15, 30, 31, 37, 43, 57, 60, 73, 85, 94, 100, 118, 121, 127, 129]
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 C68, weight-6 random supports a=[13, 41, 42, 49, 61, 62] b=[13, 17, 21, 31, 54, 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 C68, 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 68 · Z-checks 68
H_X (68 checks, sparse supports)
[2, 12, 22, 24, 32, 60, 78, 85, 111, 120, 124, 128] [4, 15, 24, 27, 35, 63, 78, 80, 113, 123, 127, 131] [6, 16, 26, 28, 36, 64, 82, 89, 115, 124, 128, 132] [7, 17, 27, 29, 37, 65, 80, 83, 106, 125, 129, 133] [8, 19, 28, 31, 39, 66, 82, 84, 117, 127, 131, 134] [9, 10, 22, 29, 30, 58, 83, 85, 108, 118, 122, 126] [1, 10, 20, 30, 32, 40, 69, 86, 93, 119, 128, 132] [11, 21, 31, 33, 41, 67, 84, 87, 110, 129, 133, 135] [3, 12, 23, 32, 35, 43, 71, 86, 88, 121, 131, 134] [13, 14, 26, 33, 34, 62, 87, 89, 112, 122, 126, 130] [4, 14, 24, 34, 36, 44, 69, 72, 90, 97, 123, 132] [5, 15, 25, 35, 37, 45, 73, 88, 91, 114, 133, 135] [7, 16, 27, 36, 39, 47, 71, 75, 90, 92, 125, 134] [0, 17, 18, 30, 37, 38, 68, 91, 93, 116, 126, 130] [8, 18, 28, 38, 40, 48, 69, 72, 76, 94, 101, 127] [9, 19, 29, 39, 41, 49, 73, 77, 92, 95, 118, 135] [11, 20, 31, 40, 43, 51, 71, 75, 79, 94, 96, 129] [2, 21, 22, 34, 41, 42, 68, 70, 95, 97, 120, 130] [12, 22, 32, 42, 44, 52, 72, 76, 80, 98, 105, 131] [13, 23, 33, 43, 45, 53, 73, 77, 81, 96, 99, 122] [15, 24, 35, 44, 47, 55, 75, 79, 83, 98, 100, 133] [6, 25, 26, 38, 45, 46, 68, 70, 74, 99, 101, 124] [16, 26, 36, 46, 48, 56, 76, 80, 84, 102, 109, 134] [17, 27, 37, 47, 49, 57, 77, 81, 85, 100, 103, 126] [19, 28, 39, 48, 51, 59, 79, 83, 87, 102, 104, 135] [10, 29, 30, 42, 49, 50, 70, 74, 78, 103, 105, 128] [20, 30, 40, 50, 52, 60, 71, 80, 84, 88, 106, 113] [21, 31, 41, 51, 53, 61, 81, 85, 89, 104, 107, 130] [23, 32, 43, 52, 55, 63, 73, 83, 87, 91, 106, 108] [14, 33, 34, 46, 53, 54, 74, 78, 82, 107, 109, 132] [24, 34, 44, 54, 56, 64, 75, 84, 88, 92, 110, 117] [25, 35, 45, 55, 57, 65, 68, 85, 89, 93, 108, 111] [27, 36, 47, 56, 59, 66, 77, 87, 91, 95, 110, 112] [18, 37, 38, 50, 57, 58, 69, 78, 82, 86, 111, 113] [1, 28, 38, 48, 58, 60, 79, 88, 92, 96, 114, 121] [29, 39, 49, 59, 61, 67, 70, 89, 93, 97, 112, 115] [3, 31, 40, 51, 60, 63, 81, 91, 95, 99, 114, 116] [22, 41, 42, 54, 61, 62, 72, 82, 86, 90, 115, 117] [4, 32, 42, 52, 62, 64, 83, 92, 96, 100, 118, 125] [5, 33, 43, 53, 63, 65, 74, 93, 97, 101, 116, 119] [7, 35, 44, 55, 64, 66, 85, 95, 99, 103, 118, 120] [0, 26, 45, 46, 58, 65, 76, 86, 90, 94, 119, 121] [0, 1, 8, 36, 46, 56, 87, 96, 100, 104, 122, 129] [9, 37, 47, 57, 66, 67, 78, 97, 101, 105, 120, 123] [1, 3, 11, 39, 48, 59, 89, 99, 103, 107, 122, 124] [2, 30, 49, 50, 62, 67, 80, 90, 94, 98, 123, 125] [2, 4, 12, 40, 50, 60, 91, 100, 104, 108, 126, 133] [3, 5, 13, 41, 51, 61, 82, 101, 105, 109, 124, 127] [4, 7, 15, 43, 52, 63, 93, 103, 107, 111, 126, 128] [0, 5, 6, 34, 53, 54, 84, 94, 98, 102, 127, 129] [6, 8, 16, 44, 54, 64, 95, 104, 108, 112, 130, 135] [7, 9, 17, 45, 55, 65, 86, 105, 109, 113, 128, 131] [8, 11, 19, 47, 56, 66, 97, 107, 111, 115, 130, 132] [2, 9, 10, 38, 57, 58, 88, 98, 102, 106, 131, 133] [1, 10, 12, 20, 48, 58, 68, 73, 99, 108, 112, 116] [11, 13, 21, 49, 59, 67, 90, 109, 113, 117, 132, 134] [3, 12, 15, 23, 51, 60, 68, 69, 101, 111, 115, 119] [6, 13, 14, 42, 61, 62, 92, 102, 106, 110, 134, 135] [4, 14, 16, 24, 52, 62, 70, 77, 103, 112, 116, 120] [5, 15, 17, 25, 53, 63, 69, 71, 94, 113, 117, 121] [7, 16, 19, 27, 55, 64, 70, 72, 105, 115, 119, 123] [0, 10, 17, 18, 46, 65, 71, 73, 96, 106, 110, 114] [0, 8, 18, 20, 28, 56, 74, 81, 107, 116, 120, 124] [9, 19, 21, 29, 57, 66, 72, 75, 98, 117, 121, 125] [1, 11, 20, 23, 31, 59, 74, 76, 109, 119, 123, 127] [2, 14, 21, 22, 50, 67, 75, 77, 100, 110, 114, 118] [3, 13, 23, 25, 33, 61, 76, 79, 102, 121, 125, 129] [5, 6, 18, 25, 26, 54, 79, 81, 104, 114, 118, 122]
H_Z (68 checks, sparse supports)
[13, 17, 21, 31, 54, 56, 81, 109, 110, 117, 129, 130] [6, 10, 14, 33, 56, 59, 74, 102, 110, 112, 122, 132] [17, 21, 25, 35, 58, 60, 68, 85, 113, 114, 121, 133] [8, 12, 16, 26, 59, 61, 76, 104, 112, 115, 124, 134] [10, 14, 18, 37, 60, 63, 69, 78, 106, 114, 116, 126] [11, 15, 19, 28, 54, 61, 79, 107, 115, 117, 127, 135] [21, 25, 29, 39, 62, 64, 70, 89, 117, 118, 125, 135] [12, 16, 20, 30, 63, 65, 71, 80, 108, 116, 119, 128] [14, 18, 22, 41, 64, 66, 72, 82, 110, 118, 120, 130] [15, 19, 23, 32, 58, 65, 73, 83, 111, 119, 121, 131] [0, 1, 25, 29, 33, 43, 73, 74, 93, 121, 122, 129] [16, 20, 24, 34, 66, 67, 75, 84, 112, 120, 123, 132] [1, 3, 18, 22, 26, 45, 68, 76, 86, 114, 122, 124] [19, 23, 27, 36, 62, 67, 77, 87, 115, 123, 125, 134] [2, 4, 29, 33, 37, 47, 77, 78, 97, 125, 126, 133] [3, 5, 20, 24, 28, 38, 69, 79, 88, 116, 124, 127] [4, 7, 22, 26, 30, 49, 70, 80, 90, 118, 126, 128] [0, 5, 23, 27, 31, 40, 71, 81, 91, 119, 127, 129] [6, 8, 33, 37, 41, 51, 81, 82, 101, 129, 130, 135] [7, 9, 24, 28, 32, 42, 72, 83, 92, 120, 128, 131] [8, 11, 26, 30, 34, 53, 74, 84, 94, 122, 130, 132] [2, 9, 27, 31, 35, 44, 75, 85, 95, 123, 131, 133] [10, 12, 37, 41, 45, 55, 68, 73, 85, 86, 105, 133] [11, 13, 28, 32, 36, 46, 76, 87, 96, 124, 132, 134] [12, 15, 30, 34, 38, 57, 68, 69, 78, 88, 98, 126] [6, 13, 31, 35, 39, 48, 79, 89, 99, 127, 134, 135] [14, 16, 41, 45, 49, 59, 70, 77, 89, 90, 109, 135] [15, 17, 32, 36, 40, 50, 69, 71, 80, 91, 100, 128] [16, 19, 34, 38, 42, 61, 70, 72, 82, 92, 102, 130] [10, 17, 35, 39, 43, 52, 71, 73, 83, 93, 103, 131] [18, 20, 45, 49, 53, 63, 73, 74, 81, 93, 94, 113] [19, 21, 36, 40, 44, 54, 72, 75, 84, 95, 104, 132] [20, 23, 38, 42, 46, 65, 68, 74, 76, 86, 96, 106] [14, 21, 39, 43, 47, 56, 75, 77, 87, 97, 107, 134] [22, 24, 49, 53, 57, 66, 77, 78, 85, 97, 98, 117] [23, 25, 40, 44, 48, 58, 69, 76, 79, 88, 99, 108] [24, 27, 42, 46, 50, 67, 70, 78, 80, 90, 100, 110] [18, 25, 43, 47, 51, 60, 71, 79, 81, 91, 101, 111] [3, 26, 28, 53, 57, 61, 81, 82, 89, 101, 102, 121] [27, 29, 44, 48, 52, 62, 72, 80, 83, 92, 103, 112] [5, 28, 31, 46, 50, 54, 74, 82, 84, 94, 104, 114] [22, 29, 47, 51, 55, 64, 75, 83, 85, 95, 105, 115] [7, 30, 32, 57, 61, 65, 85, 86, 93, 105, 106, 125] [0, 31, 33, 48, 52, 56, 76, 84, 87, 96, 107, 116] [9, 32, 35, 50, 54, 58, 78, 86, 88, 98, 108, 118] [1, 26, 33, 51, 55, 59, 79, 87, 89, 99, 109, 119] [11, 34, 36, 61, 65, 67, 89, 90, 97, 109, 110, 129] [2, 35, 37, 52, 56, 60, 80, 88, 91, 100, 111, 120] [13, 36, 39, 54, 58, 62, 82, 90, 92, 102, 112, 122] [4, 30, 37, 55, 59, 63, 83, 91, 93, 103, 113, 123] [5, 15, 38, 40, 65, 67, 93, 94, 101, 113, 114, 133] [6, 39, 41, 56, 60, 64, 84, 92, 95, 104, 115, 124] [0, 17, 40, 43, 58, 62, 86, 94, 96, 106, 116, 126] [8, 34, 41, 59, 63, 66, 87, 95, 97, 107, 117, 127] [5, 9, 19, 42, 44, 67, 97, 98, 105, 117, 118, 135] [1, 10, 43, 45, 60, 64, 88, 96, 99, 108, 119, 128] [0, 2, 21, 44, 47, 62, 90, 98, 100, 110, 120, 130] [3, 12, 38, 45, 63, 66, 91, 99, 101, 111, 121, 131] [5, 9, 13, 23, 46, 48, 73, 101, 102, 109, 121, 122] [1, 4, 14, 47, 49, 64, 92, 100, 103, 112, 123, 132] [0, 2, 6, 25, 48, 51, 68, 94, 102, 104, 114, 124] [3, 7, 16, 42, 49, 66, 95, 103, 105, 115, 125, 134] [9, 13, 17, 27, 50, 52, 77, 105, 106, 113, 125, 126] [1, 4, 8, 18, 51, 53, 69, 96, 104, 107, 116, 127] [2, 6, 10, 29, 52, 55, 70, 98, 106, 108, 118, 128] [3, 7, 11, 20, 46, 53, 71, 99, 107, 109, 119, 129] [4, 8, 12, 22, 55, 57, 72, 100, 108, 111, 120, 131] [7, 11, 15, 24, 50, 57, 75, 103, 111, 113, 123, 133]
raw submission JSON