  A novel construction of lattices is proposed. This construction can be
thought of as Construction A with codes that can be represented as the
Cartesian product of $L$ linear codes over
$\mathbb{F}_{p_1},\ldots,\mathbb{F}_{p_L}$, respectively; hence, is referred to
as the product construction. The existence of a sequence of such lattices that
are good for quantization and Poltyrev-good under multistage decoding is shown.
This family of lattices is then used to generate a sequence of nested lattice
codes which allows one to achieve the same computation rate of Nazer and
Gastpar for compute-and-forward under multistage decoding, which is referred to
as lattice-based multistage compute-and-forward.
  Motivated by the proposed lattice codes, two families of signal
constellations are then proposed for the separation-based compute-and-forward
framework proposed by Tunali \textit{et al.} together with a multilevel
coding/multistage decoding scheme tailored specifically for these
constellations. This scheme is termed separation-based multistage
compute-and-forward and is shown having a complexity of the channel coding
dominated by the greatest common divisor of the constellation size (may not be
a prime number) instead of the constellation size itself.
