Improvement on Asymptotic Density of Packing Families Derived from Multiplicative Lattices

Abstract

Let $\omega =(-1+\sqrt{-3})/2$. For any lattice $P\subseteq {\mathbb{Z}}^n$, $\mathcal{P}=P+\omega P$ is a subgroup of ${\mathcal{O}}_K^n$, where ${\mathcal{O}}_K=\mathbb{Z}[\omega ]\subseteq \mathbb{C}$. As $\mathbb{C}$ is naturally isomorphic to ${\mathbb{R}}^2$, $\mathcal{P}$ can be regarded as a lattice in ${\mathbb{R}}^{2n}$. Let $P$ be a multiplicative lattice (principal lattice or congruence lattice) introduced by Rosenbloom and Tsfasman. We concatenate a family of special codes with $t_{\mathfrak{P}}^{\ell }\cdot (P+\omega P)$, where $t_{\mathfrak{P}}$ is the generator of a prime ideal $\mathfrak{P}$ of ${\mathcal{O}}_K$. Applying this concatenation to a family of principal lattices, we obtain a new family with asymptotic density exponent $\lambda ⩾-1.26532182283$, which is better than $-1.87$ given by Rosenbloom and Tsfasman considering only principal lattice families. For a new family based on congruence lattices, the result is $\lambda ⩾-1.26532181404$, which is better than $-1.39$ by considering only congruence lattice families.