Lattices and Spherical Codes

  • Sueli I. R. Costa
  • Frédérique Oggier
  • Antonio Campello
  • Jean-Claude Belfiore
  • Emanuele Viterbo
Chapter
Part of the SpringerBriefs in Mathematics book series (BRIEFSMATH)

Abstract

Lattices in \(\mathbb {R}^{n}\) with sublattices which have an orthogonal basis are associated with spherical codes in \(\mathbb {R}^{2n}\) generated by a finite commutative group of orthogonal matrices. They also can be used to construct homogeneous spherical curves for transmitting a continuous alphabet source over an AWGN channel. In both cases, the performance of the decoding process is related to the packing density of the lattices (see ( 2.13)). In the continuous case, the packing density of these curves relies on the search for projection lattices with good packing density. We present here a survey on this topic mainly based on [18, 31, 96, 105].

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Copyright information

© The Author(s) 2017

Authors and Affiliations

  • Sueli I. R. Costa
    • 1
  • Frédérique Oggier
    • 2
  • Antonio Campello
    • 3
  • Jean-Claude Belfiore
    • 4
  • Emanuele Viterbo
    • 5
  1. 1.Institute of Mathematics, Statistics and Computer ScienceUniversity of CampinasCampinasBrazil
  2. 2.Division of Mathematical Sciences, School of Physical and Mathematical SciencesNanyang Technological UniversitySingaporeSingapore
  3. 3.Department of Electrical and Electronic EngineeringImperial College LondonLondonUK
  4. 4.Communications and Electronics DepartmentTélécom ParisTechParisFrance
  5. 5.Department of Electrical and Computer Systems EngineeringMonash UniversityClaytonAustralia

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