Improvement on Asymptotic Density of Packing Families Derived from Multiplicative Lattices

TL;DR

This paper improves the asymptotic density bounds of packing families derived from multiplicative lattices by applying concatenation with special codes, surpassing previous bounds for principal and congruence lattice families.

Contribution

It introduces a novel concatenation method with special codes to enhance the asymptotic density bounds of lattice-based packing families.

Findings

Improved density exponent for principal lattice families to ≥ -1.26532182283.

Enhanced density exponent for congruence lattice families to ≥ -1.26532181404.

Achieved better bounds than previous studies by Rosenbloom and Tsfasman.

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 $\lambda\geqslant-1.26532182283$, 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.