Dense Packings from Algebraic Number Fields and Codes

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${}_{256}$.