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