Finding the Minimum Norm and Center Density of Cyclic Lattices via Nonlinear Systems

TL;DR

This paper explores conditions simplifying the norm calculation of cyclic lattices, identifies systems of nonlinear equations to generate dense lattices in various dimensions, and compares their densities to known lattice types.

Contribution

It introduces specific conditions that simplify the analysis of cyclic lattices and constructs nonlinear systems to find dense lattices in odd and certain even dimensions.

Findings

Lattices as dense as D_n in odd dimensions

Lattices denser than A_n in certain even dimensions

Simplified norm expressions for specific cyclic lattices

Abstract

Lattices with a circulant generator matrix represent a subclass of cyclic lattices. This subclass can be described by a basis containing a vector and its circular shifts. In this paper, we present certain conditions under which the norm expression of an arbitrary vector of this type of lattice is substantially simplified, and then investigate some of the lattices obtained under these conditions. We exhibit systems of nonlinear equations whose solutions yield lattices as dense as $D_n$ in odd dimensions. As far as even dimensions, we obtain lattices denser than $A_n$ as long as $n \in 2\mathbb{Z} \backslash 4\mathbb{Z}$.