Connections between Construction D and related constructions of lattices

Abstract

Most practical constructions of lattice codes with high coding gains are multilevel constructions where each level corresponds to an underlying code component. Construction D, Construction D and Forney's code formula are classical constructions that produce such lattices explicitly from a family of nested binary linear codes. In this paper, we investigate these three closely related constructions along with the recently developed Construction Aof lattices from codes over the polynomial ring F2[u]/ua. We show that Construction by Code Formula produces a lattice packing if and only if the nested codes being used are closed under Schur product, thus proving the similarity of Construction D and Construction by Code Formula when applied to Reed-Muller codes. In addition, we relate Construction by Code Formula to Construction Aby finding a correspondence between nested binary codes and codes over F2[u]/ua. This proves that any lattice constructible using Construction by Code Formula is also constructible using Construction A. Finally, we show that Construction Aproduces a lattice if and only if the corresponding code over F2[u]/uais closed under shifted Schur product. © 2014 Springer Science+Business Media New York.