Improvement on Asymptotic Density of Packing Families Derived from Multiplicative Lattices

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 $\lambda\geqslant-1.26532182283$, 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 $\lambda\geqslant -1.26532181404$, which is better than $-1.39$ by considering only congruence lattice families.