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Connections between Construction D and related constructions of lattices

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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 \(\mathbb {F}_2[u]/u^a\). 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 \(\mathbb {F}_2[u]/u^a\). 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 \(\mathbb {F}_2[u]/u^a\) is closed under shifted Schur product.

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Notes

  1. This construction was earlier used by the name Construction \(\overline{\text{ D }}\) in [8].

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Acknowledgments

The authors would like to thank Cong Ling and Jagadeesh Harshan for their helpful comments and suggestions. The research of W. Kositwattanarerk and F. Oggier for this work is supported by the Singapore National Research Foundation under Research Grant NRF-RF2009-07.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Mahidol University, Bangkok, Thailand

    Wittawat Kositwattanarerk

  2. Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    Frédérique Oggier

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  1. Wittawat Kositwattanarerk
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  2. Frédérique Oggier
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Correspondence to Wittawat Kositwattanarerk.

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This is one of several papers published in Designs, Codes and Cryptography comprising the “Special Issue on Coding and Cryptography”.

The work of W. Kositwattanarerk was conducted in part while the author was at the Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore. The material in this paper was presented in part at the International Workshop on Coding and Cryptography, Bergen, Norway, April 2013.

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Kositwattanarerk, W., Oggier, F. Connections between Construction D and related constructions of lattices. Des. Codes Cryptogr. 73, 441–455 (2014). https://doi.org/10.1007/s10623-014-9939-3

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  • DOI: https://doi.org/10.1007/s10623-014-9939-3

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