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

Abstract

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 $Λ_{D}$, $Λ_{D'}$ and $Λ_{\overline{D}}$ and, under certain conditions, a generator matrix for $Λ_{D'}$ 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 $Λ_{D}$ and $Λ_{\overline{D}}$ attached to the distances of the codes used in these constructions.