On Lattice Constructions D and D' from q-ary Linear Codes
Franciele do Carmo Silva, Ana Paula de Souza, Eleonesio Strey, Sueli I. R. Costa
TL;DR
The paper investigates Constructions D and D' of lattices from nested $q$-ary linear codes, linking them to Construction A and deriving explicit volume and distance expressions. It provides generator- and check-matrix representations, establishes volume bounds and equality conditions, and yields $L_P$-distance formulas, including for the dual-code-based view via Construction A duality. A key contribution is extending multistage decoding with re-encoding (Encoding Method B) to $q$-ary codes in Construction D', along with a practical decoding framework that uses per-level decoders and a structured check matrix. The results illuminate how code structure and zero-one addition closures impact coding gain and center density, offering design guidance for efficient, high-performance lattice codes in AWGN and related settings. The work also identifies open questions on decoding complexity and how generator choices influence lattice parameters, motivating further exploration of decoding strategies for general $q$-ary lattices.
Abstract
Multilevel lattice codes, such as those associated to Constructions $C$, $\overline{D}$, D and D', have relevant applications in communications. In this paper, we investigate some properties of lattices obtained via Constructions D and D' from $q$-ary linear codes. Connections with Construction A, generator matrices, expressions and bounds for the lattice volume and minimum distances are derived. Extensions of previous results regarding construction and decoding of binary and $p$-ary linear codes ($p$ prime) are also presented.