On shaping complex lattice constellations from multi-level constructions

Constructions of lattices are known to have a strong connection to the study of classical linear codes over F2; one such celebrated construction is that of Barnes-Wall (BW) lattices over ℤ[i], with i = √-1, wherein weighted sum of nested Reed-Muller codes over powers of the base 1 + i leads to the famous multi-level construction of BW lattices. Although these constructions facilitate simple encoding and decoding of information bits, the resulting codewords need to be mapped onto their representatives in order to reduce the average energy of the lattice code. Drawing inspirations from the case of BW lattices, in this work, we address a general question of how to arrive at representatives of the codewords of a lattice code over ℤ[θ], where θ is a quadratic integer, that is generated via a multi-level construction over linear codes over Fq, for q > 2. In particular, we introduce a novel shaping function τ : ℤ[θ] → ℤ[θ] on the components of lattice codewords, and prove that the natural constellation of lattices from multi-level constructions can be rearranged into a multi-dimensional cube or parallelepiped under such a map. We demonstrate numerically that our mapping results in a reduction of the average energy of lattice constellations. Our proposed mapping has applications in communications, particularly in encoding and decoding of lattice codes from multi-level constructions over q-ary linear codes.