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Algebraic lattices via polynomial rings

  • Agnaldo José Ferrari
  • Antonio Aparecido de AndradeEmail author
Article
  • 29 Downloads

Abstract

Signal constellations having lattice structure have been studied as meaningful means for signal transmission over Gaussian channel. Usually the problem of finding good signal constellations for a Gaussian channel is associated with the search for lattices with high packing density, where in general the packing density is usually hard to estimate. The aim of this paper was to illustrate the fact that the polynomial ring \(\mathbb {Z}[x]\) can produce lattices with maximum achievable center density, where \(\mathbb {Z}\) is the ring of rational integers. Essentially, the method consists of constructing a generator matrix from a quotient ring of \(\mathbb {Z}[x]\).

Keywords

Galois ring Lattice Packing density Center density 

Mathematics Subject Classification

11H06 11R80 97N70 

Notes

Acknowledgements

The authors would like to thank the anonymous reviewers for their insightful comments that greatly improved the quality of this work. This work was partially supported by Fapesp 2013/25977-7 and 2014/14449-2, and CNPq 429346/2018-2.

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Copyright information

© SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2019

Authors and Affiliations

  • Agnaldo José Ferrari
    • 1
  • Antonio Aparecido de Andrade
    • 2
    Email author
  1. 1.School of SciencesSão Paulo State University (Unesp)BauruBrazil
  2. 2.Department of MathematicsSão Paulo State University (Unesp)São José do Rio PretoBrazil

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