Connections between Construction D and related constructions of lattices

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 $$\hbox {D}'$$D′, 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 $$\hbox {A}'$$A′ of lattices from codes over the polynomial ring $$\mathbb {F}_2[u]/u^a$$F2[u]/ua. 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 $$\hbox {A}'$$A′ by finding a correspondence between nested binary codes and codes over $$\mathbb {F}_2[u]/u^a$$F2[u]/ua. This proves that any lattice constructible using Construction by Code Formula is also constructible using Construction $$\hbox {A}'$$A′. Finally, we show that Construction $$\hbox {A}'$$A′ produces a lattice if and only if the corresponding code over $$\mathbb {F}_2[u]/u^a$$F2[u]/ua is closed under shifted Schur product.