Dense packings from algebraic number fields and codes

We introduce a new method from number fields and codes to construct dense packings in the Euclidean spaces. Via the canonical Q-embedding of arbitrary number field K into R[K:Q], both the prime ideal p and its residue field κ can be embedded as discrete subsets in R[K:Q]. Thus we can concatenate the embedding image of the Cartesian product of n copies of p together with the image of a length n code over κ. This concatenation leads to a packing in the Euclidean space Rn[K:Q]. Moreover, we extend the single concatenation to multiple concatenations to obtain dense packings and asymptotically good packing families. For instance, with the help of Magma, we construct a 256-dimensional packing denser than the Barnes-Wall lattice BW256.