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arXiv · 2609.22887

New Lattices with Unexpected Shallow Deep Holes

Yiming Li·Chuanming Zong
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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 $\sqrt{7/3}$, $\sqrt2$, $\sqrt{5/2}$ and $\sqrt{8/3}$, 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.

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Yiming Li, Chuanming Zong. 2026-09-19. New Lattices with Unexpected Shallow Deep Holes. https://arxiv.org/abs/2609.22887

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