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 $\piA$ (previously called product construction) recently proposed by the authors. We then discuss two generalizations where the first one is a general construction of lattices named Construction $\piD$ 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.