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[[120,6,16]] d ≤
n
120
k
6
d
16
kd²/n
12.8
w
12

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Distance

d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[6, 17, 18, 24, 31, 37, 55, 59, 65, 68, 75, 79, 82, 84, 95, 97]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[0, 12, 29, 30, 32, 39, 48, 53, 72, 76, 80, 84, 94, 98, 108, 117]
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 C60, weight-6 random supports a=[20, 22, 33, 34, 43, 47] b=[15, 32, 36, 44, 45, 55]
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 C60, 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 60 · Z-checks 60
H_X (60 checks, sparse supports)
[3, 11, 39, 42, 45, 53, 73, 78, 87, 90, 103, 110] [6, 16, 20, 47, 50, 56, 69, 79, 84, 92, 96, 108] [9, 20, 22, 35, 48, 53, 61, 63, 71, 83, 97, 116] [11, 22, 50, 52, 54, 58, 84, 90, 99, 102, 113, 117] [12, 23, 26, 32, 42, 53, 74, 85, 90, 91, 98, 103] [14, 26, 28, 30, 41, 56, 64, 66, 76, 78, 89, 102] [16, 28, 32, 55, 57, 59, 80, 91, 96, 104, 107, 116] [17, 29, 33, 38, 47, 56, 68, 81, 96, 97, 105, 108] [18, 29, 30, 32, 35, 45, 65, 69, 77, 80, 93, 101] [20, 32, 34, 46, 56, 58, 66, 71, 82, 95, 108, 119] [21, 22, 33, 35, 36, 42, 67, 71, 72, 83, 84, 107] [8, 10, 19, 22, 34, 57, 75, 96, 102, 110, 112, 118] [23, 35, 38, 44, 52, 58, 86, 97, 102, 103, 109, 113] [11, 24, 36, 38, 41, 50, 63, 70, 74, 82, 86, 99] [26, 38, 40, 42, 51, 59, 72, 76, 88, 90, 101, 112] [27, 28, 39, 41, 43, 47, 60, 76, 77, 89, 91, 113] [13, 15, 25, 28, 40, 44, 62, 92, 103, 107, 115, 119] [29, 41, 45, 49, 55, 59, 78, 93, 107, 108, 114, 116] [30, 41, 42, 44, 46, 54, 74, 80, 89, 92, 105, 111] [16, 22, 31, 32, 43, 45, 66, 75, 80, 81, 88, 93] [0, 4, 19, 32, 44, 59, 76, 82, 85, 94, 106, 116] [33, 34, 45, 46, 47, 52, 77, 82, 83, 95, 96, 115] [0, 15, 18, 21, 31, 34, 68, 79, 87, 107, 112, 117] [2, 8, 19, 35, 46, 49, 65, 98, 108, 112, 113, 118] [22, 36, 47, 49, 51, 57, 71, 81, 86, 94, 98, 110] [23, 28, 37, 38, 48, 50, 62, 72, 86, 87, 95, 99] [1, 7, 25, 38, 49, 52, 68, 83, 88, 100, 102, 111] [39, 40, 50, 51, 53, 55, 63, 88, 89, 101, 103, 118] [1, 2, 24, 27, 37, 40, 69, 73, 85, 104, 113, 115] [5, 13, 25, 41, 51, 54, 70, 90, 105, 115, 116, 119] [2, 4, 10, 42, 51, 52, 67, 86, 92, 101, 104, 114] [28, 34, 43, 44, 53, 54, 76, 87, 92, 93, 100, 105] [2, 3, 12, 25, 31, 44, 60, 64, 88, 94, 97, 119] [0, 4, 8, 45, 54, 55, 73, 89, 94, 95, 106, 107] [0, 3, 27, 30, 33, 43, 68, 75, 78, 91, 99, 115] [4, 5, 9, 18, 31, 46, 68, 74, 79, 109, 116, 118] [5, 7, 15, 34, 47, 55, 60, 82, 93, 98, 109, 117] [35, 40, 48, 49, 56, 57, 69, 83, 98, 99, 106, 110] [5, 6, 8, 17, 37, 49, 61, 67, 78, 95, 100, 112] [1, 7, 13, 50, 57, 58, 71, 79, 100, 101, 111, 113] [1, 6, 9, 36, 39, 48, 62, 73, 80, 84, 97, 118] [7, 10, 14, 24, 37, 51, 73, 81, 85, 102, 114, 119] [7, 8, 9, 12, 21, 52, 62, 70, 77, 98, 104, 111] [0, 2, 10, 40, 53, 58, 61, 88, 99, 104, 105, 114] [2, 9, 11, 23, 37, 43, 60, 63, 72, 85, 100, 108] [3, 10, 12, 13, 18, 54, 60, 64, 84, 101, 106, 115] [4, 12, 14, 20, 30, 43, 65, 78, 79, 86, 91, 119] [0, 13, 14, 17, 27, 55, 63, 65, 75, 94, 105, 109] [1, 5, 15, 46, 56, 59, 64, 80, 95, 109, 110, 117] [5, 14, 16, 18, 29, 48, 61, 66, 68, 77, 90, 106] [6, 15, 17, 19, 24, 57, 61, 67, 82, 91, 111, 118] [7, 17, 21, 26, 36, 48, 70, 84, 85, 93, 97, 112] [8, 17, 18, 20, 23, 33, 62, 67, 69, 81, 89, 109] [1, 3, 9, 19, 21, 58, 62, 66, 70, 100, 110, 114] [10, 11, 21, 23, 24, 30, 63, 64, 72, 73, 96, 111] [3, 13, 24, 26, 29, 39, 60, 65, 71, 74, 87, 114] [4, 6, 14, 25, 27, 59, 65, 72, 75, 92, 106, 117] [15, 16, 27, 29, 31, 36, 66, 67, 77, 79, 94, 103] [6, 11, 19, 20, 31, 33, 61, 69, 70, 76, 81, 117] [12, 16, 25, 26, 37, 39, 64, 74, 75, 83, 87, 104]
H_Z (60 checks, sparse supports)
[15, 32, 36, 44, 45, 55, 80, 82, 93, 94, 103, 107] [2, 38, 43, 49, 50, 58, 86, 88, 99, 100, 108, 113] [16, 25, 40, 42, 52, 53, 83, 88, 90, 92, 103, 104] [2, 13, 27, 44, 47, 54, 60, 92, 94, 105, 113, 115] [5, 32, 45, 48, 54, 59, 80, 90, 93, 95, 106, 116] [8, 23, 46, 47, 55, 56, 89, 95, 96, 98, 108, 109] [5, 9, 19, 49, 53, 57, 61, 98, 100, 110, 116, 118] [10, 30, 38, 50, 52, 57, 86, 96, 99, 101, 102, 111] [7, 22, 26, 34, 35, 49, 71, 83, 93, 98, 102, 112] [1, 8, 28, 37, 52, 58, 62, 95, 100, 102, 104, 113] [13, 29, 42, 51, 53, 58, 71, 90, 101, 103, 105, 114] [2, 9, 10, 24, 39, 55, 60, 63, 73, 104, 114, 118] [10, 14, 25, 44, 54, 56, 64, 92, 102, 105, 106, 119] [0, 28, 33, 40, 41, 54, 76, 89, 99, 105, 107, 115] [4, 13, 18, 35, 55, 59, 65, 101, 106, 107, 109, 116] [11, 19, 34, 47, 56, 59, 76, 82, 96, 108, 110, 117] [5, 14, 15, 20, 31, 58, 61, 66, 79, 109, 117, 119] [8, 15, 21, 42, 49, 57, 67, 98, 107, 110, 111, 112] [0, 5, 17, 34, 38, 46, 68, 82, 95, 105, 109, 112] [1, 22, 35, 39, 46, 57, 71, 80, 83, 110, 113, 118] [6, 8, 18, 19, 40, 48, 61, 62, 69, 106, 112, 118] [7, 19, 24, 41, 52, 58, 70, 82, 102, 111, 113, 114] [9, 13, 20, 21, 36, 50, 62, 63, 70, 71, 79, 84] [2, 10, 21, 26, 37, 59, 64, 72, 85, 104, 112, 114] [1, 3, 10, 40, 45, 51, 73, 88, 101, 110, 114, 115] [4, 20, 28, 41, 44, 51, 76, 86, 89, 92, 116, 119] [12, 13, 24, 25, 30, 46, 64, 65, 74, 111, 115, 119] [0, 22, 25, 31, 55, 59, 75, 88, 94, 107, 116, 117] [14, 19, 26, 27, 32, 43, 65, 66, 75, 76, 85, 91] [5, 15, 18, 27, 33, 52, 67, 68, 77, 109, 115, 117] [0, 3, 4, 14, 29, 49, 65, 68, 78, 94, 106, 114] [4, 6, 15, 34, 46, 50, 79, 82, 92, 95, 117, 118] [1, 16, 18, 30, 31, 56, 64, 66, 68, 69, 79, 80] [8, 17, 19, 31, 36, 51, 67, 70, 81, 94, 112, 118] [20, 24, 32, 33, 47, 57, 69, 71, 81, 82, 91, 96] [9, 21, 25, 33, 38, 48, 62, 68, 70, 72, 83, 97] [1, 6, 7, 11, 21, 54, 70, 73, 84, 100, 111, 117] [2, 7, 12, 32, 40, 51, 85, 88, 98, 101, 104, 119] [4, 23, 24, 36, 37, 42, 67, 72, 73, 74, 85, 86] [3, 13, 25, 34, 37, 43, 60, 75, 87, 100, 115, 119] [26, 31, 38, 39, 44, 53, 74, 76, 87, 88, 97, 103] [8, 14, 27, 30, 39, 45, 65, 73, 75, 77, 78, 89] [3, 5, 11, 12, 26, 41, 60, 64, 70, 74, 78, 90] [0, 4, 12, 16, 27, 57, 75, 79, 91, 94, 104, 106] [6, 28, 30, 42, 43, 59, 72, 76, 78, 80, 91, 92] [7, 18, 29, 31, 43, 47, 60, 68, 77, 79, 81, 93] [20, 33, 37, 45, 49, 56, 69, 78, 81, 83, 95, 108] [6, 10, 16, 17, 22, 33, 61, 67, 75, 81, 84, 96] [1, 7, 9, 17, 23, 44, 62, 85, 97, 100, 109, 111] [12, 35, 36, 47, 48, 52, 77, 83, 84, 86, 97, 98] [0, 11, 24, 37, 48, 53, 61, 63, 73, 85, 87, 99] [18, 26, 39, 42, 50, 54, 74, 84, 87, 89, 90, 101] [11, 14, 22, 23, 38, 51, 63, 72, 81, 86, 90, 102] [3, 12, 15, 23, 28, 39, 60, 62, 64, 87, 91, 103] [17, 30, 41, 43, 53, 55, 63, 78, 89, 91, 93, 105] [16, 21, 28, 29, 34, 45, 66, 77, 87, 93, 96, 107] [2, 6, 17, 20, 29, 35, 61, 65, 67, 69, 97, 108] [3, 22, 36, 48, 56, 58, 66, 71, 84, 97, 99, 110] [11, 23, 27, 35, 40, 50, 63, 69, 72, 99, 103, 113] [9, 16, 29, 32, 41, 46, 66, 74, 77, 80, 108, 116]
raw submission JSON