2013/08/26 by Wittawat Kositwattanarerk, Kositwattanarerk, Wittawat, Frédérique Oggier +1
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #FOS: Computer and information sciences #Information Theory (cs.IT) #cs.IT #graph theory and CDMA systems #math.IT #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1308.6175
Submitted to Designs, Codes and Cryptography
openalex publication_date 2013/08/26 · arxiv created 2014/01/02 · arxiv updated 2014/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 \mathbbF2[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 A' by finding a correspondence between nested binary codes and codes over \mathbbF2[u]/ua. 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 \mathbbF2[u]/ua is closed under shifted Schur product.