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[[126,6,16]] d ≤
n
126
k
6
d
16
kd²/n
12.19
w
12

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Distance

d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[1, 3, 4, 6, 8, 15, 23, 30, 59, 67, 71, 82, 97, 112, 118, 124]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[0, 11, 22, 26, 27, 29, 39, 59, 60, 66, 84, 88, 90, 93, 99, 124]
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 on C21xC3, weight-6 random supports a=[4,17,24,41,48,50] b=[10,23,39,48,52,56]
model Xiaomi MiMo-V2.5 (claimed, not verified)
builds on arXiv:2306.16400
date 2026-07-28
notes Found by GAP-enhanced autoresearch (see fieldnotes/2026-07-28-gap-campaign-results.md). 2BGA code on C21xC3, weight-6 random supports. Distance upper bound confirmed flat across 400->2k->8k->60k->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 63 · Z-checks 63
H_X (63 checks, sparse supports)
[1, 11, 20, 26, 37, 43, 75, 78, 91, 100, 101, 109] [3, 4, 9, 25, 32, 49, 79, 81, 84, 88, 89, 97] [5, 19, 28, 34, 45, 51, 64, 66, 99, 108, 109, 116] [6, 20, 29, 35, 46, 52, 84, 87, 100, 109, 110, 117] [0, 6, 14, 31, 38, 54, 70, 85, 90, 94, 95, 103] [8, 10, 16, 33, 40, 56, 67, 69, 87, 96, 97, 105] [9, 11, 17, 34, 41, 57, 88, 90, 93, 97, 98, 106] [12, 27, 29, 35, 36, 42, 68, 71, 83, 92, 101, 116] [13, 28, 37, 43, 53, 58, 69, 72, 108, 116, 117, 122] [2, 14, 29, 38, 44, 54, 93, 96, 109, 117, 118, 123] [2, 13, 22, 39, 46, 59, 63, 74, 93, 102, 103, 111] [3, 14, 23, 40, 47, 60, 78, 94, 99, 103, 104, 112] [15, 17, 18, 24, 41, 48, 72, 73, 76, 80, 89, 105] [16, 19, 25, 42, 49, 61, 74, 77, 96, 105, 106, 114] [5, 17, 20, 26, 43, 50, 97, 99, 102, 106, 107, 115] [21, 36, 38, 44, 45, 51, 76, 79, 92, 101, 110, 122] [7, 22, 37, 46, 52, 59, 77, 80, 116, 122, 123, 125] [0, 8, 23, 38, 47, 60, 67, 73, 102, 105, 117, 123] [7, 21, 23, 30, 47, 53, 65, 77, 82, 86, 95, 111] [8, 22, 31, 48, 54, 62, 66, 83, 102, 111, 112, 119] [9, 10, 23, 32, 49, 55, 87, 103, 108, 112, 113, 120] [24, 26, 27, 33, 50, 56, 80, 82, 85, 89, 98, 114] [12, 25, 28, 34, 51, 57, 83, 86, 105, 114, 115, 121] [1, 13, 26, 29, 35, 52, 63, 65, 106, 108, 111, 115] [0, 30, 45, 47, 53, 58, 85, 88, 101, 110, 118, 125] [2, 15, 31, 46, 54, 62, 73, 81, 86, 89, 122, 125] [3, 10, 16, 32, 47, 55, 73, 74, 82, 111, 114, 123] [15, 30, 32, 39, 55, 59, 71, 86, 91, 95, 104, 119] [16, 18, 31, 40, 56, 60, 72, 92, 111, 119, 120, 124] [4, 17, 19, 32, 41, 57, 64, 68, 96, 112, 116, 120] [1, 33, 35, 36, 42, 61, 89, 91, 94, 98, 107, 121] [5, 21, 34, 37, 43, 58, 65, 70, 92, 95, 114, 121] [2, 6, 22, 35, 38, 44, 65, 66, 71, 115, 116, 119] [3, 7, 39, 53, 55, 59, 67, 81, 94, 97, 110, 118] [8, 18, 24, 40, 54, 60, 81, 82, 90, 95, 98, 125] [4, 9, 19, 25, 41, 55, 73, 82, 83, 91, 119, 121] [4, 24, 39, 41, 48, 62, 79, 95, 100, 104, 113, 124] [10, 25, 27, 40, 49, 61, 68, 75, 80, 101, 119, 124] [5, 11, 26, 28, 41, 50, 68, 69, 76, 105, 120, 122] [6, 12, 42, 44, 45, 51, 63, 70, 98, 100, 103, 107] [7, 13, 30, 43, 46, 52, 70, 71, 78, 101, 104, 121] [0, 8, 14, 31, 44, 47, 65, 71, 72, 79, 122, 124] [4, 9, 15, 48, 59, 62, 67, 74, 90, 103, 106, 118] [10, 16, 27, 33, 49, 60, 81, 90, 91, 99, 104, 107] [4, 11, 17, 28, 34, 50, 70, 82, 91, 92, 100, 124] [11, 18, 33, 48, 50, 56, 64, 75, 88, 104, 109, 113] [12, 19, 34, 36, 49, 57, 75, 76, 84, 89, 110, 124] [1, 13, 20, 35, 37, 50, 68, 76, 77, 85, 114, 125] [0, 14, 21, 51, 53, 58, 63, 66, 78, 107, 109, 112] [2, 15, 22, 39, 52, 54, 70, 78, 79, 87, 110, 113] [0, 3, 16, 23, 40, 55, 71, 75, 79, 80, 88, 125] [11, 17, 18, 24, 56, 62, 67, 74, 83, 99, 112, 115] [10, 19, 25, 36, 42, 57, 63, 90, 99, 100, 108, 113] [1, 20, 27, 42, 56, 61, 64, 69, 84, 97, 113, 117] [5, 21, 28, 43, 45, 57, 75, 84, 85, 93, 98, 118] [1, 6, 22, 29, 44, 46, 76, 81, 85, 86, 94, 121] [3, 7, 23, 30, 58, 59, 63, 66, 72, 87, 117, 120] [2, 8, 24, 31, 48, 60, 64, 78, 87, 88, 96, 118] [18, 20, 26, 27, 33, 61, 65, 74, 83, 92, 108, 120] [6, 12, 29, 36, 51, 61, 64, 69, 77, 93, 106, 123] [5, 13, 30, 37, 52, 53, 67, 84, 93, 94, 102, 107] [7, 9, 15, 32, 39, 62, 66, 68, 72, 80, 96, 123] [12, 14, 21, 38, 45, 58, 69, 73, 77, 86, 102, 115]
H_Z (63 checks, sparse supports)
[10, 23, 39, 48, 52, 56, 67, 80, 87, 104, 111, 113] [2, 29, 45, 53, 57, 59, 63, 86, 93, 110, 116, 118] [18, 23, 31, 32, 41, 58, 72, 73, 88, 95, 112, 120] [2, 19, 32, 48, 56, 61, 64, 74, 89, 96, 113, 119] [5, 17, 33, 42, 51, 60, 64, 92, 98, 99, 105, 107] [7, 29, 37, 38, 47, 61, 65, 77, 94, 101, 117, 123] [5, 8, 38, 53, 59, 62, 66, 67, 95, 102, 118, 122] [4, 31, 39, 40, 44, 49, 79, 81, 96, 103, 119, 124] [7, 27, 32, 40, 41, 50, 68, 80, 82, 97, 104, 120] [8, 12, 28, 41, 56, 61, 64, 69, 83, 98, 105, 124] [12, 17, 25, 26, 35, 62, 68, 83, 89, 100, 106, 115] [10, 13, 26, 42, 51, 58, 63, 69, 101, 107, 108, 114] [0, 37, 45, 46, 50, 54, 70, 85, 102, 109, 122, 125] [12, 15, 38, 46, 47, 55, 71, 73, 86, 103, 110, 123] [13, 16, 18, 47, 59, 62, 67, 72, 74, 104, 111, 125] [0, 11, 40, 48, 49, 57, 75, 88, 90, 105, 112, 124] [1, 15, 36, 41, 49, 50, 68, 76, 89, 91, 106, 113] [12, 16, 21, 37, 50, 61, 69, 75, 77, 92, 107, 114] [1, 25, 33, 34, 43, 55, 75, 91, 97, 108, 114, 121] [18, 21, 26, 34, 35, 44, 65, 76, 92, 98, 109, 115] [7, 19, 22, 35, 51, 58, 63, 66, 77, 110, 116, 121] [1, 3, 46, 53, 54, 60, 78, 81, 94, 111, 117, 125] [4, 21, 24, 47, 54, 55, 73, 79, 82, 95, 112, 118] [18, 22, 25, 27, 55, 62, 74, 80, 81, 83, 113, 119] [3, 5, 20, 49, 56, 57, 75, 84, 97, 99, 114, 120] [1, 6, 24, 45, 50, 57, 64, 76, 85, 98, 100, 115] [1, 12, 21, 25, 30, 46, 63, 77, 84, 86, 101, 121] [4, 6, 34, 42, 43, 52, 70, 84, 100, 106, 116, 121] [0, 27, 30, 35, 43, 44, 65, 71, 85, 101, 107, 117] [7, 15, 28, 31, 44, 58, 66, 70, 72, 86, 118, 122] [6, 9, 10, 54, 59, 60, 81, 87, 90, 103, 119, 123] [4, 11, 30, 33, 55, 60, 67, 82, 88, 91, 104, 120] [4, 18, 27, 31, 34, 36, 64, 83, 89, 90, 92, 124] [5, 9, 13, 29, 57, 61, 68, 84, 93, 106, 108, 121] [1, 5, 6, 14, 33, 53, 65, 69, 85, 94, 107, 109] [6, 21, 30, 34, 39, 54, 66, 70, 86, 93, 95, 110] [2, 11, 14, 43, 51, 52, 70, 78, 93, 109, 115, 122] [0, 3, 36, 39, 44, 52, 63, 71, 79, 94, 110, 123] [0, 7, 15, 24, 37, 40, 67, 72, 78, 80, 95, 125] [10, 14, 17, 19, 60, 62, 73, 90, 96, 99, 112, 124] [4, 10, 11, 20, 39, 42, 68, 74, 91, 97, 100, 113] [11, 27, 36, 40, 43, 45, 69, 75, 92, 98, 99, 101] [5, 12, 13, 17, 22, 38, 70, 76, 93, 102, 115, 116] [6, 13, 14, 23, 42, 59, 63, 71, 77, 94, 103, 117] [14, 30, 39, 43, 48, 60, 72, 78, 95, 102, 104, 118] [2, 8, 20, 23, 52, 58, 65, 78, 87, 102, 117, 125] [0, 2, 3, 9, 45, 48, 66, 73, 79, 88, 103, 118] [3, 15, 24, 33, 46, 49, 74, 80, 81, 87, 89, 104] [10, 18, 19, 23, 26, 28, 75, 82, 99, 105, 108, 120] [11, 19, 20, 29, 48, 51, 64, 76, 83, 100, 106, 109] [20, 36, 45, 49, 52, 53, 77, 84, 101, 107, 108, 110] [13, 21, 22, 26, 31, 47, 65, 78, 85, 102, 111, 122] [14, 22, 23, 32, 51, 62, 66, 79, 86, 103, 112, 123] [2, 7, 8, 16, 29, 32, 71, 81, 87, 96, 111, 123] [3, 8, 9, 17, 53, 56, 67, 72, 82, 88, 97, 112] [9, 24, 33, 42, 54, 57, 83, 89, 90, 96, 98, 113] [19, 27, 28, 32, 35, 37, 68, 84, 91, 108, 114, 116] [20, 28, 29, 38, 56, 58, 69, 85, 92, 109, 115, 117] [22, 30, 31, 35, 40, 55, 71, 87, 94, 111, 119, 125] [8, 15, 16, 25, 38, 41, 73, 79, 90, 96, 105, 119] [9, 16, 17, 26, 59, 61, 74, 80, 91, 97, 106, 120] [28, 36, 37, 41, 44, 46, 76, 93, 100, 116, 121, 122] [16, 24, 25, 34, 47, 50, 82, 88, 99, 105, 114, 124]
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