On construction D and related constructions of lattices from linear codes

• Main Authors: Kositwattanarerk, Wittawat, Oggier, Frederique
• Other Authors: School of Physical and Mathematical Sciences
• Format: Conference or Workshop Item
• Language: English
• Published: 2013
• Subjects: DRNTU::Engineering::Civil engineering::Construction technology
• Online Access: https://hdl.handle.net/10356/95282; http://hdl.handle.net/10220/10104; http://www.selmer.uib.no/WCC2013/
• Institution: Nanyang Technological University

We investigate three closely related constructions of lattices from linear codes: the classical Construction D, Construction D ̅, and the recently developed Construction A’. These constructions have been proven useful and result in efficient encoding and decoding algorithms for Barnes-Wall lattices. Here we analyze their applications in a general setting. We show that Construction D ̅ 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 D ̅ when applied to Reed-Muller codes. In addition, we provide a correspondence between nested binary linear codes and codes over polynomial rings. This proves that Construction A’ does not always produce a lattice, but any lattices constructible using Construction D ̅ are also constructible using Construction A’. This result also gives a partial condition for Construction A’ to produce a lattice.