Construction π_A and π_D Lattices: Construction, Goodness, and Decoding Algorithms

A novel construction of lattices is proposed. This construction can be thought of as a special class of Construction A from codes over finite rings that can be represented as the Cartesian product of L linear codes over F-p1 , ... , F-pL, respectively, and hence is referred to as Construction pi(A). The existence of a sequence of such lattices that is good for channel coding (i.e., Poltyrev-limit achieving) under multistage decoding is shown. A new family of multilevel nested lattice codes based on Construction pi(A) lattices is proposed and its achievable rate for the additive white Gaussian noise channel is analyzed. A generalization named Construction pi(D) is also investigated, which subsumes Construction A with codes over prime fields, Construction D, and Construction pi(A) as special cases.