Connections between Construction D and related constructions of lattices |
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Authors: | Wittawat Kositwattanarerk Frédérique Oggier |
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Institution: | 1. Department of Mathematics, Faculty of Science, Mahidol University, Bangkok, Thailand 2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore
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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 \(\hbox {D}'\) , 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 \(\hbox {A}'\) of lattices from codes over the polynomial ring \(\mathbb {F}_2u]/u^a\) . 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 \(\hbox {A}'\) by finding a correspondence between nested binary codes and codes over \(\mathbb {F}_2u]/u^a\) . This proves that any lattice constructible using Construction by Code Formula is also constructible using Construction \(\hbox {A}'\) . Finally, we show that Construction \(\hbox {A}'\) produces a lattice if and only if the corresponding code over \(\mathbb {F}_2u]/u^a\) is closed under shifted Schur product. |
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