Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8

by Achill Schürmann; Frank Vallentin


Preprint series: 04-29, Preprints

11H31 Lattice packing and covering, See also {05B40, 52C15, 52C17}
05B40 Packing and covering, See also {11H31, 52C15, 52C17}
52C17 Packing and covering in $n$ dimensions, See also {05B40, 11H31}


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

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