N ov 2 00 4 Local Covering Optimality of Lattices : Leech Lattice versus Root Lattice
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چکیده
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 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.
منابع مشابه
. M G ] 1 5 O ct 2 00 4 Local Covering Optimality of Lattices : Leech Lattice versus Root Lattice
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 form...
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We show the highly non–surprising fact that the Leech lattice gives a sphere covering which is locally least dense among lattice coverings. This gives a first example of a locally optimal lattice covering having a non–simplicial Delone subdivision. Hereby, we in particular answer a question of Dickson posed in 1968. By showing that the Leech lattice is rigid, our answer is even strongest possib...
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