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

Multistage Compute-and-Forward with Multilevel Lattice Codes Based on Product Constructions