On the density of cyclotomic lattices constructed from codes

TL;DR

This paper constructs families of cyclotomic lattices using code-based methods to achieve improved lower bounds on lattice sphere packing densities, extending recent theoretical results with explicit examples.

Contribution

It provides an effective construction of lattices reaching the improved density bounds using codes over cyclotomic fields and Construction A.

Findings

Explicit families of lattices attain the improved density bounds.

The construction leverages codes over cyclotomic fields.

Results apply to infinitely many dimensions.

Abstract

Recently, Venkatesh improved the best known lower bound for lattice sphere packings by a factor $\log\log n$ for infinitely many dimensions $n$. Here we prove an effective version of this result, in the sense that we exhibit, for the same set of dimensions, finite families of lattices containing a lattice reaching this bound. Our construction uses codes over cyclotomic fields, lifted to lattices via Construction A.