Dense generic well-rounded lattices

Abstract

It is well-known that the densest lattice sphere packings also typically have large kissing numbers. The sphere packing density maximization problem is known to have a solution among well-rounded lattices, of which the integer lattice $\mathbb{Z}^n$ is the simplest example. The integer lattice is also an example of a generic well-rounded lattice, i.e., a well-rounded lattice with a minimal kissing number. However, the integer lattice has the worst density among well-rounded lattices. In this paper, the problem of constructing explicit generic well-rounded lattices with dense sphere packings is considered. To this end, so-called tame lattices recently introduced by Damir and Mantilla-Soler are utilized. Tame lattices came to be as a generalization of the ring of integers of certain abelian number fields. The sublattices of tame lattices constructed in this paper are shown to always result in either a generic well-rounded lattice or the lattice $A_n$, with density ranging between that of $\mathbb{Z}^n$ and $A_n$. In order to find generic well-rounded lattices with densities beyond that of $A_n$, explicit deformations of some known densest lattice packings are constructed, yielding a family of generic well-rounded lattices with densities arbitrarily close to the optimum. In addition to being an interesting mathematical problem on its own right, the constructions are also motivated from a more practical point of view. Namely, generic well-rounded lattices with high packing density make good candidates for lattice codes used in secure wireless communications.

Figures (1)

• Figure 1: Center density of the deformed hexagonal lattice $\Lambda_h^\alpha$ as a function of $\alpha$. The bottom line shows $\delta(\mathbb{Z}^2)$ and the upper line $\delta(\Lambda_h)$.

Theorems & Definitions (63)

• Definition 1
• Example 1
• Example 2
• Definition 2
• Remark 1
• Definition 3
• Definition 4
• Lemma 1
• proof
• Remark 2