Connections Between Construction D and Related Constructions of Lattices

Abstract: Most practical constructions of lattice codes with high coding gains are multilevel constructions where each level corresponds to an underlying code component. Construction D, Construction D$&#39;$, and Forney's code formula are classical constructions that produce such lattices explicitly from a family of nested binary linear codes. In this paper, we investigate these three closely related constructions along with the recently developed Construction A$&#39;$ of lattices from codes over the polynomial ring $\mathbb{F}_2[u]/u<sup>a$. We show that Construction by Code Formula produces a lattice packing if and only if the nested codes being used are closed under Schur product, thus proving the similarity of Construction D and Construction by Code Formula when applied to Reed-Muller codes. In addition, we relate Construction by Code Formula to Construction A$&#39;$ by finding a correspondence between nested binary codes and codes over $\mathbb{F}_2[u]/u<sup>a$. This proves that any lattice constructible using Construction by Code Formula is also constructible using Construction A$&#39;$. Finally, we show that Construction A$&#39;$ produces a lattice if and only if the corresponding code over $\mathbb{F}_2[u]/u<sup>a$ is closed under shifted Schur product.