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[[128,12,16]] d ≤
n
128
k
12
d
16
kd²/n
24.0
w
16

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Distance

d_X 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[8, 13, 14, 20, 41, 43, 48, 58, 64, 65, 74, 76, 105, 106, 118, 121]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness operator (support, 16 qubits)
[4, 5, 11, 24, 26, 27, 32, 40, 76, 82, 87, 93, 104, 110, 111, 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 C8xQ8, weight-8 supports a=[4, 7, 14, 16, 21, 23, 36, 60] b=[9, 10, 18, 45, 48, 50, 58, 63]
model Xiaomi MiMo-V2.5 (claimed, not verified)
builds on arXiv:2306.16400
date 2026-07-29
notes Found by GAP-enhanced autoresearch sweep 3 (weight-8). 2BGA code on C8xQ8. Distance 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

the research note submitted with this code · raw markdown · all notes

Note for [[128,12,16]]

Direction and hypothesis

This run targeted the weight-9plus / unrestricted board cell with GAP-based 2BGA search over small-order groups, using the same screening funnel that had already produced several other candidate families. The hypothesis was that weight-8 supports on non-abelian groups could produce a code that beats the current frontier in this cell without requiring a more structured local layout.

What was searched

The candidate was generated from the GAP-based 2BGA sweep over orders 60-120 with support weight 8, using random supports and the repo's screening pipeline. The sweep used the updated checkpoint/resume path so previously processed fingerprints were skipped, and the candidate was filtered through the standard surrogate-distance screening with the usual minimum thresholds for k and d.

Evidence trail

The candidate was screened as a promising hit and then validated with the repository's trusted gate. The validator reported:

  • verify: passed
  • refute: no lighter logical found in 7620 RIS trials
  • novelty: advances the weight-9plus × unrestricted board

Dead ends

The search also surfaced nearby weight-8 candidates that did not advance the board. In particular, the closely related [[128,20,14]] candidate was rejected by the validator as an exact duplicate of the existing board entry and did not advance its board cell. This is a useful reminder that strong-looking surrogate scores are not sufficient on their own.

Tools

  • GAP-based search pipeline
  • gf2_fast screening path
  • verify/validate_candidate.py for the trusted gate

Reproduction

The packaged code is stored at codes/128-12-16.json.

References

  • GAP4: The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.12.2; 2023. (https://www.gap-system.org)

Parity checks

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