Abstract
A novel construction of lattices is proposed. This construction can be thought of as a special class of Construction A from codes over finite rings that can be represented as the Cartesian product of $L$ linear codes over $\mathbb {F}_{p_{1}},\ldots ,\mathbb {F}_{p_{L}}$ , respectively, and hence is referred to as Construction $\pi _{A}$ . The existence of a sequence of such lattices that is good for channel coding (i.e., Poltyrev-limit achieving) under multistage decoding is shown. A new family of multilevel nested lattice codes based on Construction $\pi _{A}$ lattices is proposed and its achievable rate for the additive white Gaussian noise channel is analyzed. A generalization named Construction $\pi _{D}$ is also investigated, which subsumes Construction A with codes over prime fields, Construction D, and Construction $\pi _{A}$ as special cases.
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