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

Lattices have been used in several problems in coding theory and cryptography. In this paper we approach $q$-ary lattices obtained via Constructions D, ${\text{D}}^{\mathrm{\prime }}$ and $\bar{D}$. It is shown connections between Constructions D and ${\text{D}}^{\mathrm{\prime }}$. Bounds for the minimum $l_1$-distance of lattices ${\mathrm{\Lambda }}_D$, ${\mathrm{\Lambda }}_{D^{\mathrm{\prime }}}$ and ${\mathrm{\Lambda }}_{\bar{D}}$ and, under certain conditions, a generator matrix for ${\mathrm{\Lambda }}_{D^{\mathrm{\prime }}}$ 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 ${\mathrm{\Lambda }}_D$ and ${\mathrm{\Lambda }}_{\bar{D}}$ attached to the distances of the codes used in these constructions.