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 $\sqrt{7/3}$, $\sqrt{2}$, $\sqrt{5/2}$ and $\sqrt{8/3}$, given by $E_7^{\ast }$, $E_8$, the laminated lattices ${\Lambda }_9$ and ${\Lambda }_{10}$, 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 $\Lambda /2\Lambda$ in $7,8,9,10$ and $12$ dimensions.

15 pages