Connections Between Construction D and Related Constructions of Lattices

TL;DR

This paper explores the relationships between classical lattice constructions (D, D', Forney's formula) and the newer Construction A' over polynomial rings, establishing conditions for when these methods produce lattices and how they relate.

Contribution

It demonstrates the equivalence of Construction D and Construction by Code Formula under certain conditions and relates Construction A' to nested binary codes, unifying different lattice construction methods.

Findings

Construction by Code Formula yields a lattice if nested codes are Schur product closed.

Construction A' produces a lattice if the code over polynomial ring is shifted Schur product closed.

Any lattice from Construction by Code Formula can also be constructed via Construction A'.

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 A$'$ of lattices from codes over the polynomial ring $\mathbb{F}_2[u]/u^a$. 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 A$'$ by finding a correspondence between nested binary codes and codes over $\mathbb{F}_2[u]/u^a$. This proves that any lattice constructible using Construction by Code Formula is also constructible using Construction A$'$. Finally, we show that Construction A$'$ produces a lattice if and only if the corresponding code over $\mathbb{F}_2[u]/u^a$ is closed under shifted Schur product.