Computational Approaches to Lattice Packing and Covering Problems

Discrete Comput Geom 35:73116 (2006) Discrete & ComputationalDOI: 10.1007/s00454-005-1202-2Geometry 2005 Springer Science+Business Media, Inc.Computational Approaches to Lattice Packing

and Covering ProblemsAchill Sch

urmann1 and Frank Vallentin21Department of Mathematics, University of Magdeburg,

39106 Magdeburg, Germany

[email protected] Institute of Mathematics, The Hebrew University of Jerusalem,

Jerusalem 91904, [email protected]. We 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 Voronois reduction theory for positive definite quadratic forms, which describes all

possible Delone triangulations of Zd.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.1. OverviewTwo classical problems in the geometry of numbers are the determination of the most

economical lattice sphere packings and coverings of the Euclidean d-space Ed. In this The work by Frank Vallentin was partially supported by the Edmund Landau Center for Research in

Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). Both authors

were supported by the Deutsche Forschungsgemeinschaft (DFG) under Grant SCHU 1503/4-1.74 A. Sch

urmann and F. Vallentinpaper we describe algorithms for the lattice covering and the simultaneous lattice packing

and covering problem (lattice packing-covering problem in what follows).Roughly speaking, both problems are concerned with the most economical way to

cover Ed. In the case of the lattice covering problem, the goal is to maximize the volume

of a fundamental domain in a lattice covering with unit spheres. Roughly speaking,

we want to minimize the number of unit spheres which are needed to cover arbitrarily

large but finite regions of Ed. The objective of the lattice packing-covering problem is

to...