On Construction D and Related Constructions of Lattices from Linear Codes
نویسندگان
چکیده
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.
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