Connections between Construction D and related constructions of lattices

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• 作者：Wittawat Kositwattanarerk (1)
  Frédérique Oggier (2)
  
• 关键词：Lattices ; Lattices from codes ; Coset codes ; Barnes–Wall lattices ; Schur product of codes ; 94B05 ; 06B99
• 刊名：Designs, Codes and Cryptography
• 出版年：2014
• 出版时间：November 2014
• 年：2014
• 卷：73
• 期：2
• 页码：441-455
• 全文大小：274 KB
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  4. Forney G.D.: Coset codes-part II: binary lattices and related codes. IEEE Trans. Inf. Theory 34(5), 1152-187 (1988).
  5. Forney G.D., Trott M.D., Chung S.-Y.: Sphere-bound-achieving coset codes and multilevel coset codes. IEEE Trans. Inf. Theory 46(3), 820-50 (2000).
  6. Harshan J., Viterbo E., Belfiore J.-C.: Construction of Barnes–Wall lattices from linear codes over rings. In: Proceedings of the IEEE International Symposium on Information Theory, Cambridge, MA, , 1- July 2012, pp. 3110-114 (2012).
  7. Harshan J., Viterbo E., Belfiore J.-C.: Practical encoders and decoders for Euclidean codes from Barnes–Wall lattices. http://arxiv.org/abs/1203.3282v2. Mar 2012.
  8. Kositwattanarerk W., Oggier F.: On Construction D and related constructions of lattices from linear codes. In: Proceedings of the International Workshop on Coding and Cryptography, Bergen, Norway, 15-9 April 2013, pp. 428-37 (2013).
  9. Oggier F., Solé P., Belfiore J.-C.: Lattice codes for the wiretap Gaussian channel: construction and analysis. http://arxiv.org/abs/1103.4086.
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  12. Yan Y., Ling C., Wu X.: Polar lattices: where Arikan meets Forney. In: Proceedings of IEEE International Symposium on Information Theory, Istanbul, Turkey, 7-2 July 2013, pp. 1292-296 2013.
  
• 作者单位：Wittawat Kositwattanarerk (1)
  Frédérique Oggier (2)
  
  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
  
• ISSN：1573-7586

文摘

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.