Computational Approaches to Lattice Packing and Covering Problems

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dc.creatorSchuermann, Achill
dc.creatorVallentin, Frank
dc.date2004-03-16
dc.date2005-07-27
dc.date.accessioned2026-07-07T06:24:53Z
dc.date.available2026-07-07T06:24:53Z
dc.descriptionWe describe algorithms which address two classical problems in lattice geometry: the lattice covering and the simultaneous lattice packing-covering problem. Theoretically our algorithms solve the two problems in any fixed dimension d in the sense that they approximate optimal covering lattices and optimal packing-covering lattices within any desired accuracy. Both algorithms involve semidefinite programming and are based on Voronoi's reduction theory for positive definite quadratic forms, which describes all possible Delone triangulations of Z^d. In practice, our implementations reproduce known results in dimensions d <= 5 and in particular solve the two problems in these dimensions. For d = 6 our computations produce new best known covering as well as packing-covering lattices, which are closely related to the lattice (E6)*. For d = 7, 8 our approach leads to new best known covering lattices. Although we use numerical methods, we made some effort to transform numerical evidences into rigorous proofs. We provide rigorous error bounds and prove that some of the new lattices are locally optimal.
dc.description(v3) 40 pages, 5 figures, 6 tables, some corrections, accepted in Discrete and Computational Geometry, see also http://fma2.math.uni-magdeburg.de/~latgeo/
dc.identifierhttps://arxiv.org/abs/math/0403272
dc.identifierhttp://arxiv.org/abs/math/0403272
dc.identifierDiscrete and Computational Geometry 35 (2006), 73-116
dc.identifierdoi:10.1007/s00454-005-1202-2
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/96697
dc.subjectMetric Geometry
dc.subjectCombinatorics
dc.subject11H31
dc.titleComputational Approaches to Lattice Packing and Covering Problems
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