PreprintarXiv licencearXiv
New Lattices with Unexpected Shallow Deep Holes
Abstract
Among lattices normalized to have packing radius one, the known shallowest depths of the deep holes in $7$-, $8$-, $9$- and $10$-dimensional lattices are √(7/3), √2, √(5/2) and √(8/3), given by E7*, E8, the laminated lattices Λ9 and Λ10, respectively. This paper presents new lattices in these dimensions with shallower deep holes. It also answers two problems of J. Martinet about short representatives of the classes of Λ/2Λ in $7,8,9,10$ and $12$ dimensions.
Related
- Monadic Second-Order Logic in HOL: Deep and Shallow with Automated Faithfulness (Extended Preprint)
- Deep-water and shallow-water limits of statistical equilibria for the intermediate long wave equation
- Deep holes of full-length Reed--Solomon codes
- Covering Radius and Deep Holes of Hyperderivative Reed-Solomon Codes under the NRT Metric
- Deep holes of full-length Reed--Solomon codes: uniform bounds and exact classifications
BibTeX
@misc{li2026lattices,
title = {{New Lattices with Unexpected Shallow Deep Holes}},
author = {Yiming Li and Chuanming Zong},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22887},
eprint = {2609.22887},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22887},
}
RIS
TY - UNPB
TI - New Lattices with Unexpected Shallow Deep Holes
AU - Yiming Li
AU - Chuanming Zong
PY - 2026
DA - 2026/09/19
PB - arXiv
DO - 10.48550/arxiv.2609.22887
UR - https://arxiv.org/abs/2609.22887
AB - Among lattices normalized to have packing radius one, the known shallowest depths of the deep holes in $7$-, $8$-, $9$- and $10$-dimensional lattices are √(7/3), √2, √(5/2) and √(8/3), given by E_7^*, E_8, the laminated lattices Λ_9 and Λ_10, respectively. This paper presents new lattices in these dimensions with shallower deep holes. It also answers two problems of J. Martinet about short representatives of the classes of Λ/2Λ in $7,8,9,10$ and $12$ dimensions.
ER -