  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 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.
