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In this paper, we extend the lattice Constructions $D$, $D'$ and$\overline{D}$ $($this latter is also known as Forney's code formula$)$ fromcodes over $\mathbb{F}_p$ to linear codes over $\mathbb{Z}_q$, where $q \in\mathbb{N}$. We define an operation in $\mathbb{Z}_q^n$ called zero-oneaddition, which coincides with the Schur product when restricted to$\mathbb{Z}_2^n$ and show that the extended Construction $\overline{D}$produces a lattice if and only if the nested codes are closed under thisaddition. A generalization to the real case of the recently developedConstruction $A'$ is also derived and we show that this construction produces alattice if and only if the corresponding code over $\mathbb{Z}_q[X]/X^a$ isclosed under a shifted zero-one addition. One of the motivations for this workis the recent use of $q$-ary lattices in cryptography.

Lattices from codes over $\mathbb{Z}_q$: Generalization of Constructions $D$, $D'$ and $\overline{D}$