  Lattices have been used in several problems in coding theory and
cryptography. In this paper we approach $q$-ary lattices obtained via
Constructions D, $\D'$ and $\overline{D}$. It is shown connections between
Constructions D and $\D'$. Bounds for the minimum $l_1$-distance of lattices
$\Lambda_{D}$, $\Lambda_{D'}$ and $\Lambda_{\overline{D}}$ and, under certain
conditions, a generator matrix for $\Lambda_{D'}$ are presented. In addition,
when the chain of codes used is closed under the zero-one addition, we derive
explicit expressions for the minimum $l_1$-distances of the lattices
$\Lambda_{D}$ and $\Lambda_{\overline{D}}$ attached to the distances of the
codes used in these constructions.
