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Methods in the Local Theory of Packing and Covering Lattices

by Achill Schürmann, Frank Vallentin.

Series: 2004-34, Preprints

11H31 Lattice packing and covering
05B40 Packing and covering
52C17 Packing and covering in $n$ dimensions

In this paper we are concerned with three lattice problems: the lattice packing problem, the lattice covering problem and the lattice packing-covering problem. One way to find optimal lattices for these problems is to enumerate all finitely many, locally optimal lattices. For the lattice packing problem there are two classical algorithms going back to Minkowski and Voronoi. For the covering and for the packing-covering problem we propose new algorithms. Here we give a brief survey about these approaches. We report on some recent computer based computations where we were able to reproduce and partially extend the known classification of locally optimal lattices. Furthermore we found new record breaking covering and packing-covering lattices. We describe several methods with examples to show that a lattice is a locally optimal solution to one of the three problems.

lattice, covering, packing, Minkowski, Voronoi, local optima