In this paper, we extend the lattice Constructions D, D′ and D¯ (this latter is also known as Forney’s code formula) from codes over Fp to linear codes over Zq, where q∈N. We define an operation in Znq called zero-one addition, which coincides with the Schur product when restricted to Zn2 and show that the extended Construction D¯ produces a lattice if and only if the nested codes are closed under this addition. A generalization to the real case of the recently developed Construction A′ is also derived and we show that this construction produces a lattice if and only if the corresponding code over Zq[X]/Xa is closed under a shifted zero-one addition. One of the motivations for this work is the recent use of q-ary lattices in cryptography. Ver menos