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

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