On construction D and related constructions of lattices from linear codes
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...
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| Main Authors: | , |
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| Other Authors: | |
| Format: | Conference or Workshop Item |
| Language: | English |
| Published: |
2013
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/95282 http://hdl.handle.net/10220/10104 http://www.selmer.uib.no/WCC2013/ |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | 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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