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$_{256}$.
- Publication:
-
arXiv e-prints
- Pub Date:
- June 2015
- DOI:
- 10.48550/arXiv.1506.00419
- arXiv:
- arXiv:1506.00419
- Bibcode:
- 2015arXiv150600419C
- Keywords:
-
- Mathematics - Number Theory;
- Computer Science - Information Theory;
- Mathematics - Commutative Algebra;
- Mathematics - Metric Geometry;
- 52C17;
- 11R04;
- 94B65 (Primary);
- 94B05;
- 11H31 (Secondary)
- E-Print:
- Finite Fields and Their Applications vol. 45, 217-236 (2017)