Methods in the Local Theory of Packing and Covering Lattices

TL;DR

This paper reviews classical and new algorithms for finding locally optimal lattices in packing, covering, and packing-covering problems, reports computational results, and introduces new record-breaking lattices.

Contribution

It introduces new algorithms for covering and packing-covering lattice problems and reports computational discoveries of new optimal lattices.

Findings

Reproduced and extended known classifications of locally optimal lattices.

Discovered new record-breaking covering and packing-covering lattices.

Provided methods to verify local optimality of lattices.

Abstract

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.