Product construction is a novel lattice construction which can be thought of as Construction A with codes that can be represented as the Cartesian product of many linear codes over finite fields satisfying some conditions. 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 with a significantly less decoding complexity.