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

Achill Schuermann·Frank Vallentin
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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 E8. For this we construct a less dense covering lattice whose Delone subdivision has a common refinement with the Delone subdivision of E8. The new lattice yields a sphere covering which is more than 12% less dense than the formerly best known given by the lattice A8*. 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.

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BibTeXRIS

Achill Schuermann, Frank Vallentin. 2004-11-10. Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8. https://doi.org/10.1155/imrn.2005.1937

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