Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8
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.
Cite
@article{arxiv.math/0405441,
title = {Local Covering Optimality of Lattices: Leech Lattice versus Root Lattice E8},
author = {Achill Schuermann and Frank Vallentin},
journal= {arXiv preprint arXiv:math/0405441},
year = {2009}
}
Comments
13 pages; (v2) major revision: proof of rigidity corrected, full discussion of E8-case included, src of (v3) contains MAGMA program, (v4) some corrections