# Random Ensembles of Lattices from Generalized Reductions

## TL;DR

This paper introduces a comprehensive framework for constructing dense Euclidean lattices from linear codes, providing new density bounds, explicit constructions, and algorithmic methods that improve upon previous approaches.

## Contribution

It develops general conditions for lattice density from linear codes, applies them to number field lattices, and offers explicit, efficient constructions of dense lattices with improved parameters.

## Key findings

- Established conditions for dense lattice ensembles from linear codes.
- Derived the best known packing density for ideal lattices.
- Provided explicit, algorithmically effective lattice constructions.

## Abstract

We propose a general framework to study constructions of Euclidean lattices from linear codes over finite fields. In particular, we prove general conditions for an ensemble constructed using linear codes to contain dense lattices (i.e., with packing density comparable to the Minkowski-Hlawka lower bound). Specializing to number field lattices, we obtain a number of interesting corollaries - for instance, the best known packing density of ideal lattices, and an elementary coding-theoretic construction of asymptotically dense Hurwitz lattices. All results are algorithmically effective, in the sense that, for any dimension, a finite family containing dense lattices is exhibited. For suitable constructions based on Craig's lattices, this family is significantly smaller, in terms of alphabet-size, than previous ones in the literature.