by Achill Schürmann, Frank Vallentin.

**Series:** 2004-29, Preprints

- MSC:
- 11H31 Lattice packing and covering
- 05B40 Packing and covering
- 52C17 Packing and covering in $n$ dimensions

**Abstract:**

We show that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings. We show that a similar result is false for the root lattice $\mathsf{E}_8$. For this we construct a less dense covering lattice whose Delone subdivision has a common refinement with the Delone subdivision of $\mathsf{E}_8$. The

new lattice yields a sphere covering which is more than $12\%$ less dense than the formerly best known given by the lattice $\mathsf{A}_8^*$. Currently, the Leech lattice is the first and only known example of a locally optimal lattice covering having a non-simplicial Delone subdivision. We hereby in particular answer a question of Dickson posed in 1968. By showing that the Leech lattice is rigid our answer is even strongest possible in a sense.

**Keywords:**

lattice, covering, Leech lattice, root lattice E8