Lattices from codes for harnessing interference: An overview and generalizations

In this paper, using compute-and-forward as an example, we provide an overview of constructions of lattices from codes that possess the right algebraic structures for harnessing interference. This includes Construction A, Construction D, and Construction π_A (previously called product construction) recently proposed by the authors. While most of the results in this paper have been available in the literature, we discuss two generalizations where the first one is a general construction of lattices named Construction π_D subsuming the above three constructions as special cases and the second one is to go beyond principal ideal domains and build lattices over algebraic integers.