Dense packings from algebraic number fields and codes

作者：Cheng Shantian^*

来源：Finite Fields and Their Applications, 2017, 45: 217-236.

DOI：10.1016/j.ffa.2016.12.006

• Dense packings
• Number fields
• Minkowski lattice
• 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 kappa 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 kappa. This concatenation leads to a packing in the Euclidean space R-n[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.

• 出版日期2017-5