← back to the board
[[128,8,15]] d ≤
n
128
k
8
d
15
kd²/n
14.062
w
12

Share this result

Distance

d_X 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[11, 13, 24, 50, 52, 66, 67, 85, 88, 94, 100, 106, 111, 117, 126]
d_Z 15 · witness weight 15 (claimed upper_bound)
witness operator (support, 15 qubits)
[3, 12, 13, 19, 27, 35, 50, 51, 52, 57, 88, 100, 108, 114, 126]
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 cyclic group C64, weight-6 random supports a=[12,13,25,26,44,51] b=[4,22,32,36,38,59]
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 C64, 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 64 · Z-checks 64
H_X (64 checks, sparse supports)
[22, 29, 47, 48, 54, 55, 73, 82, 97, 101, 105, 115] [4, 35, 40, 52, 59, 60, 77, 92, 107, 111, 115, 119] [9, 41, 45, 50, 56, 57, 83, 87, 98, 104, 119, 124] [11, 23, 48, 55, 59, 62, 84, 90, 99, 118, 120, 125] [2, 11, 28, 42, 60, 62, 64, 65, 74, 103, 116, 119] [7, 12, 13, 30, 43, 49, 66, 68, 73, 93, 98, 104] [24, 33, 35, 44, 51, 61, 67, 78, 83, 93, 94, 118] [6, 13, 38, 51, 54, 61, 80, 93, 108, 114, 120, 124] [15, 16, 41, 52, 60, 63, 94, 96, 109, 123, 125, 126] [5, 7, 15, 34, 56, 63, 65, 66, 78, 113, 121, 124] [3, 19, 22, 23, 36, 53, 71, 73, 77, 99, 108, 114] [17, 20, 43, 45, 55, 62, 72, 88, 93, 99, 100, 123] [0, 25, 26, 51, 56, 57, 100, 102, 105, 106, 126, 127] [0, 3, 10, 25, 44, 61, 66, 70, 71, 88, 117, 120] [4, 8, 16, 29, 46, 58, 67, 77, 84, 87, 109, 118] [19, 27, 30, 38, 60, 63, 76, 81, 99, 105, 109, 110] [2, 12, 24, 28, 30, 63, 67, 72, 79, 85, 91, 126] [13, 22, 31, 32, 45, 49, 76, 80, 90, 100, 105, 112] [1, 35, 43, 46, 52, 61, 68, 97, 102, 112, 113, 119] [14, 23, 31, 32, 50, 57, 69, 74, 75, 77, 117, 120] [1, 8, 15, 45, 52, 58, 70, 75, 80, 95, 99, 126] [9, 25, 26, 34, 36, 59, 76, 79, 81, 84, 94, 115] [1, 6, 18, 32, 55, 61, 64, 69, 110, 112, 115, 127] [1, 8, 14, 18, 20, 62, 67, 71, 75, 81, 122, 125] [9, 12, 21, 26, 35, 39, 72, 80, 87, 94, 102, 123] [0, 29, 33, 36, 48, 56, 86, 91, 102, 103, 109, 115] [0, 7, 17, 22, 34, 36, 72, 76, 89, 95, 97, 127] [4, 23, 37, 38, 42, 53, 86, 90, 96, 110, 115, 116] [2, 19, 39, 45, 57, 62, 73, 107, 112, 116, 117, 124] [6, 16, 24, 37, 42, 54, 74, 78, 79, 87, 122, 125] [2, 12, 21, 25, 38, 57, 75, 79, 90, 101, 109, 127] [13, 18, 32, 41, 44, 46, 86, 89, 91, 94, 100, 119] [1, 3, 4, 27, 44, 46, 64, 76, 82, 86, 101, 107] [5, 9, 16, 47, 48, 58, 68, 96, 103, 106, 119, 121] [6, 7, 29, 38, 49, 63, 77, 83, 116, 120, 121, 122] [5, 11, 17, 26, 47, 59, 78, 80, 85, 88, 89, 126] [6, 7, 18, 22, 31, 48, 64, 79, 89, 96, 102, 111] [20, 23, 28, 39, 42, 51, 68, 82, 97, 100, 110, 124] [14, 25, 33, 42, 50, 51, 64, 81, 91, 92, 96, 101] [8, 9, 15, 34, 47, 57, 66, 69, 82, 92, 114, 116] [10, 26, 36, 44, 53, 60, 77, 83, 98, 103, 113, 125] [10, 27, 37, 46, 53, 63, 65, 85, 95, 96, 100, 102] [2, 8, 9, 20, 33, 39, 65, 68, 83, 86, 92, 111] [10, 13, 21, 26, 40, 52, 69, 70, 73, 106, 113, 124] [0, 3, 11, 35, 48, 53, 70, 85, 87, 93, 121, 125] [10, 15, 27, 32, 40, 41, 81, 82, 88, 90, 95, 127] [3, 4, 11, 28, 37, 52, 65, 82, 89, 104, 106, 112] [5, 16, 30, 34, 49, 55, 73, 92, 103, 107, 110, 120] [5, 18, 24, 43, 54, 55, 65, 91, 97, 98, 106, 111] [6, 12, 13, 25, 40, 44, 71, 74, 92, 98, 118, 121] [14, 20, 32, 46, 56, 58, 87, 93, 108, 113, 117, 126] [0, 14, 33, 39, 47, 58, 66, 95, 101, 106, 110, 112] [10, 17, 19, 28, 59, 60, 66, 69, 97, 104, 107, 108] [11, 18, 20, 22, 23, 50, 67, 70, 78, 98, 108, 123] [21, 24, 30, 39, 42, 61, 80, 84, 99, 117, 122, 127] [1, 21, 24, 40, 43, 49, 69, 71, 101, 103, 111, 116] [7, 27, 29, 37, 54, 58, 70, 72, 78, 104, 114, 117] [14, 27, 29, 34, 41, 56, 71, 74, 83, 84, 107, 114] [4, 15, 16, 28, 30, 54, 72, 75, 84, 88, 105, 108] [5, 17, 31, 36, 49, 62, 64, 85, 90, 94, 109, 122] [2, 17, 19, 31, 50, 53, 67, 74, 104, 111, 113, 121] [3, 21, 33, 35, 47, 59, 75, 76, 88, 114, 118, 122] [8, 31, 40, 41, 43, 45, 79, 81, 85, 86, 118, 123] [12, 19, 37, 38, 50, 51, 68, 89, 91, 95, 105, 123]
H_Z (64 checks, sparse supports)
[4, 22, 32, 36, 38, 59, 76, 77, 89, 90, 108, 115] [4, 9, 41, 42, 46, 48, 82, 84, 86, 87, 96, 119] [5, 9, 13, 39, 51, 52, 68, 80, 92, 94, 106, 124] [6, 14, 16, 23, 53, 60, 74, 77, 96, 108, 110, 125] [5, 18, 33, 37, 42, 63, 65, 78, 91, 96, 110, 122] [19, 22, 39, 43, 52, 55, 73, 97, 99, 111, 112, 123] [13, 20, 43, 44, 53, 56, 71, 86, 93, 98, 100, 113] [10, 13, 23, 49, 55, 57, 69, 73, 90, 98, 100, 120] [11, 16, 24, 26, 56, 58, 78, 84, 87, 103, 106, 126] [0, 5, 10, 28, 43, 47, 66, 85, 88, 97, 103, 106] [4, 19, 29, 49, 57, 60, 77, 104, 105, 107, 109, 116] [19, 20, 23, 30, 58, 61, 67, 68, 99, 108, 110, 117] [15, 17, 21, 26, 32, 61, 69, 80, 88, 94, 113, 127] [1, 10, 14, 19, 34, 40, 69, 71, 81, 95, 107, 113] [6, 9, 29, 35, 53, 56, 83, 87, 102, 114, 115, 121] [16, 21, 29, 30, 36, 62, 72, 73, 84, 103, 109, 122] [7, 17, 20, 24, 35, 54, 72, 78, 93, 97, 111, 122] [15, 21, 23, 38, 45, 62, 75, 90, 99, 116, 123, 124] [0, 32, 37, 39, 45, 46, 86, 87, 95, 100, 112, 117] [2, 6, 34, 40, 42, 57, 74, 79, 92, 116, 124, 127] [3, 14, 21, 54, 57, 58, 75, 87, 101, 106, 114, 117] [16, 35, 41, 44, 59, 62, 88, 94, 107, 118, 119, 125] [25, 27, 31, 32, 42, 62, 64, 74, 81, 90, 100, 117] [2, 14, 24, 29, 44, 50, 67, 74, 83, 91, 101, 117] [11, 13, 35, 45, 58, 61, 70, 80, 93, 112, 118, 119] [26, 31, 35, 36, 46, 63, 76, 77, 85, 94, 102, 113] [3, 17, 27, 30, 45, 59, 76, 85, 88, 99, 104, 107] [16, 25, 31, 38, 48, 63, 79, 96, 105, 109, 120, 121] [1, 38, 39, 42, 47, 49, 68, 80, 101, 110, 116, 122] [5, 6, 7, 11, 44, 50, 64, 78, 89, 98, 120, 121] [6, 8, 21, 24, 31, 59, 69, 79, 80, 111, 118, 122] [20, 26, 41, 45, 51, 63, 81, 83, 100, 123, 124, 126] [8, 27, 33, 36, 38, 41, 81, 83, 86, 95, 109, 114] [0, 18, 26, 37, 48, 52, 70, 89, 102, 106, 115, 125] [2, 5, 40, 48, 49, 53, 73, 85, 90, 103, 111, 121] [3, 10, 11, 15, 20, 54, 65, 70, 82, 88, 108, 125] [11, 12, 17, 31, 37, 41, 74, 85, 89, 90, 104, 123] [0, 30, 32, 38, 51, 55, 91, 93, 105, 110, 120, 127] [12, 18, 24, 25, 36, 41, 71, 79, 91, 94, 98, 127] [4, 25, 33, 40, 47, 55, 88, 92, 101, 106, 115, 118] [2, 5, 46, 52, 56, 60, 65, 107, 109, 113, 119, 126] [0, 12, 15, 17, 58, 63, 66, 72, 95, 109, 121, 126] [12, 33, 43, 46, 48, 51, 68, 91, 93, 101, 102, 118] [1, 28, 32, 47, 52, 57, 69, 75, 82, 112, 119, 126] [7, 10, 50, 52, 53, 58, 70, 77, 95, 96, 104, 113] [8, 14, 15, 25, 30, 59, 66, 75, 81, 84, 92, 126] [15, 22, 27, 37, 47, 51, 78, 82, 95, 96, 105, 114] [1, 36, 42, 48, 55, 60, 64, 97, 99, 103, 115, 125] [17, 18, 22, 28, 46, 51, 64, 67, 89, 97, 100, 108] [9, 18, 40, 43, 50, 60, 69, 81, 98, 111, 119, 123] [7, 10, 39, 56, 57, 61, 66, 83, 102, 117, 124, 127] [0, 1, 21, 22, 25, 27, 70, 71, 76, 101, 102, 127] [4, 27, 28, 34, 39, 55, 65, 72, 82, 84, 107, 110] [13, 19, 28, 50, 54, 56, 74, 91, 104, 105, 108, 124] [3, 6, 14, 49, 61, 62, 64, 71, 93, 112, 120, 122] [1, 2, 4, 18, 31, 33, 64, 67, 75, 86, 111, 112] [3, 7, 13, 19, 34, 47, 66, 73, 76, 89, 114, 121] [9, 33, 34, 44, 49, 60, 66, 76, 83, 92, 94, 103] [23, 29, 34, 54, 59, 61, 78, 84, 97, 114, 115, 120] [8, 11, 24, 53, 62, 63, 65, 67, 85, 99, 116, 125] [2, 7, 9, 28, 37, 43, 65, 68, 72, 79, 104, 116] [3, 8, 23, 29, 40, 44, 70, 71, 77, 82, 86, 118] [8, 12, 16, 20, 35, 50, 67, 68, 75, 87, 92, 123] [12, 22, 26, 30, 45, 54, 72, 73, 79, 80, 98, 105]
raw submission JSON