Lattices and Spherical Codes

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


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].


  1. 4.
    C. Alves, S.I.R. Costa, Commutative group codes in and —approaching the bound. Discret. Math. 313(16), 1677–1687 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 11.
    E. Biglieri, M. Elia, Cyclic-group codes for the gaussian channel (corresp.). IEEE Trans. Inf. Theory 22(5), 624–629 (1976)Google Scholar
  3. 16.
    G. Caire, E. Biglieri, Linear block codes over cyclic groups. IEEE Trans. Inf. Theory 41(5), 1246–1256 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 17.
    A. Campello, J. Strapasson, S.I.R. Costa, On projections of arbitrary lattices. Linear Algebra Appl. 439(9), 2577–2583 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 18.
    A. Campello, C. Torezzan, S.I.R. Costa, Curves on flat tori and analog source-channel codes. IEEE Trans. Inf. Theory 59(10), 6646–6654 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 22.
    H. Cohn, N. Elkies, New upper bounds on sphere packings I. Ann. Math. 157(2), 689–714 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 26.
    J.H. Conway, N.J.A. Sloane, Sphere-Packings, Lattices, and Groups (Springer, New York, 1998)zbMATHGoogle Scholar
  8. 27.
    S.I.R. Costa, On closed twisted curves. Proc. Am. Math. Soc. 109(1), 205–214 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 28.
    S.I.R. Costa, E. Agustini, M. Muniz, R. Palazzo, Slepian-type codes on a flat torus, in IEEE International Symposium on Information Theory (2000), p. 58Google Scholar
  10. 29.
    S.I.R. Costa, M. Muniz, E. Agustini, R. Palazzo, Graphs, tessellations, and perfect codes on flat tori. IEEE Trans. Inf. Theory 50(10), 2363–2377 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 30.
    S.I.R. Costa, J.E. Strapasson, M.M.S. Alves, T.B. Carlos, Circulant graphs and tesselations on flat tori. Linear Algebra Appl. 432, 369–382 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 31.
    S.I.R. Costa, C. Torezzan, A. Campello, V.A. Vaishampayan, Flat tori, lattices and spherical codes, in 2013 Information Theory and Applications Workshop (ITA), (2013), pp. 1–8Google Scholar
  13. 35.
    A.A. El Gamal, L.A. Hemachandra, I. Shperling, V.K. Wei, Using simulated anneling to design good codes. IEEE Trans. Inf. Theory IT-33(1), 116–123 (1987)CrossRefGoogle Scholar
  14. 37.
    T. Ericson, V. Zinoviev, Codes on Euclidean Spheres. (North-Holland Mathematical Library, Amsterdam, 2001)Google Scholar
  15. 40.
    G.D. Forney Jr., Coset codes. I. Introduction and geometrical classification. IEEE Trans. Inf. Theory 34(5), 1123–1151 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 42.
    G.D. Forney Jr., Geometrically uniform codes. IEEE Trans. Inf. Theory 37(5), 1241–1260 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 43.
    F.R. Gantmacher, The Theory of Matrices, vol. 1 (Translated from the Russian by K. A. Hirsch. Reprint of the 1959 translation). (AMS Chelsea Publishing, Providence, 1998)Google Scholar
  18. 44.
    T.J. Goblick, Theoretical limitations on the transmission of data from analog sources. IEEE Trans. Inf. Theory 11(4), 558–567 (1965)CrossRefzbMATHGoogle Scholar
  19. 47.
    J. Hamkins, K. Zeger, Asymptotically dense spherical codes. I. Wrapped spherical codes. IEEE Trans. Inf. Theory 43(6), 1774–1785 (1997)CrossRefzbMATHGoogle Scholar
  20. 48.
    J. Hamkins, K. Zeger, Asymptotically dense spherical codes II. Laminated spherical codes. IEEE Trans. Inf. Theory 43(6), 1786–1798 (1997)CrossRefzbMATHGoogle Scholar
  21. 54.
    I. Ingemarsson, Group Codes for the Gaussian Channel, in Topics in Coding Theory. Lecture Notes in Control and Information Sciences, vol.128 (Springer, Berlin, 1989), pp. 73–108Google Scholar
  22. 64.
    H.-A. Loeliger, Signal sets matched to groups. IEEE Trans. Inf. Theory 37(6), 1675–1682 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 68.
    J. Martinet, Perfect Lattices in Euclidean Spaces (Springer, Berlin, 2013)zbMATHGoogle Scholar
  24. 70.
    C.D. Meyer, Matrix Analysis and Applied Linear Algebra (Society for Industrial Mathematics (SIAM), Philadelphia, 2000)Google Scholar
  25. 84.
    W.W. Peterson, J.B. Nation, M.P. Fossorier, Reflection group codes and their decoding. IEEE Trans. Inf. Theory 56(12), 6273–6293 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 91.
    C.E. Shannon, Communication in the presence of noise. Proc. IRE 37(1), 10–21 (1949)MathSciNetCrossRefGoogle Scholar
  27. 96.
    R.M. Siqueira, S.I.R. Costa, Flat tori, lattices and bounds for commutative group codes. Des. Codes Crypt. 49(1–3), 307–321 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 97.
    D. Slepian, Group codes for the gaussian channel. Bell Syst. Tech. J. 47, 575–602 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 99.
    N.J.A. Sloane, V.A. Vaishampayan, S.I.R. Costa, A note on projecting the cubic lattice. Discret. Comput. Geom. 46(3), 472–478 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 100.
    N.J.A. Sloane, V.A. Vaishampayan, S.I.R. Costa, The lifting construction: a general solution for the fat strut problem, in IEEE International Symposium on Information Theory (2010), pp. 1037–1041Google Scholar
  31. 103.
    J. Stillwell, Geometry of Surfaces. Universitext (Springer, New York, 1992)CrossRefzbMATHGoogle Scholar
  32. 105.
    C. Torezzan, S.I.R. Costa, V.A. Vaishampayan, Constructive spherical codes on layers of flat tori. IEEE Trans. Inf. Theory 59(10), 6655–6663 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 106.
    C. Torezzan, J.E. Strapasson, S.I.R. Costa, R.M. Siqueira, Optimum commutative group codes. Des. Codes Crypt. 74(2), 379–394 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 107.
    V.A. Vaishampayan, S.I.R. Costa, Curves on a sphere, shift-map dynamics, and error control for continuous alphabet sources. IEEE Trans. Inf. Theory 49(7), 1658–1672 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 108.
    V.A. Vaishampayan, N.J.A. Sloane, S.I.R. Costa, Dynamical systems, curves and coding for continuous alphabet sources, in Proceedings of International Telecommunications Symposium, ITW2002, Bangalore (2002)Google Scholar
  36. 110.
    A. Zaghloul, R.M. Taylor Jr., L. Mili, Structured spherical codes with optimal distance distributions, in IEEE International Symposium on Information Theory (2017)Google Scholar

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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