Bounds for the $l_1$-distance of $q$-ary lattices obtained via Constructions D, D$^{}$ and $overline{D}$

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.