Dense Packings from Algebraic Number Fields and Codes

TL;DR

This paper presents a novel method using algebraic number fields and coding theory to construct dense Euclidean packings, achieving densities surpassing known lattices like Barnes-Wall in high dimensions.

Contribution

It introduces a new algebraic and coding-based framework for constructing dense packings in Euclidean spaces, including methods for asymptotically good packing families.

Findings

Constructed a 256-dimensional packing denser than Barnes-Wall lattice.

Developed a concatenation method combining number field embeddings and codes.

Extended the approach to multiple concatenations for asymptotic density improvements.

Abstract

We introduce a new method from number fields and codes to construct dense packings in the Euclidean spaces. Via the canonical $\mathbb{Q}$-embedding of arbitrary number field $K$ into $\mathbb{R}^{[K:\mathbb{Q}]}$, both the prime ideal $\mathfrak{p}$ and its residue field $\kappa$ can be embedded as discrete subsets in $\mathbb{R}^{[K:\mathbb{Q}]}$. Thus we can concatenate the embedding image of the Cartesian product of $n$ copies of $\mathfrak{p}$ together with the image of a length $n$ code over $\kappa$. This concatenation leads to a packing in Euclidean space $\mathbb{R}^{n[K:\mathbb{Q}]}$. Moreover, we extend the single concatenation to multiple concatenation to obtain dense packings and asymptotically good packing families. For instance, with the help of \Magma{}, we construct one $256$-dimension packing denser than the Barnes-Wall lattice BW$_{256}$.