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A novel construction of lattices is proposed. This construction can bethought of as a special class of Construction A from codes over finite ringsthat 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 referredto as Construction $\pi_A$. The existence of a sequence of such lattices thatis good for channel coding (i.e., Poltyrev-limit achieving) under multistagedecoding is shown. A new family of multilevel nested lattice codes based onConstruction $\pi_A$ lattices is proposed and its achievable rate for theadditive white Gaussian channel is analyzed. A generalization namedConstruction $\pi_D$ is also investigated which subsumes Construction A withcodes over prime fields, Construction D, and Construction $\pi_A$ as specialcases.

Construction $\pi_A$ and $\pi_D$ Lattices: Construction, Goodness, and Decoding Algorithms