Lattices and Applications

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


A lattice in \(\mathbb {R}^{n}\) is a set of points (vectors) composed by all integer linear combinations of independent vectors.


  1. 1.
    J.F. Adams, Lectures on Lie Groups (Midway Reprints Series) (University of Chicago Press, Chicago, 1983)Google Scholar
  2. 2.
    E. Agrell, T. Eriksson, A. Vardy, K. Zeger, Closest point search in lattices. IEEE Trans. Inf. Theory 48(8), 2201–2214 (2002)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    M. Ajtai, Generating hard instances of lattice problems, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’96, New York, NY (ACM, 1996), pp. 99–108Google Scholar
  4. 6.
    W. Banaszczyk, New bounds in some transference theorems in the geometry of numbers. Math. Ann. 296(1), 625–635 (1993)MathSciNetCrossRefMATHGoogle Scholar
  5. 8.
    E.S. Barnes, G.E. Wall, Some extreme forms defined in terms of abelian groups. J. Aust. Math. Soc. 1(1), 47–63 (1959)MathSciNetCrossRefMATHGoogle Scholar
  6. 14.
    N. Bourbaki, Lie Groups and Lie Algebras: Chapters 4–6 (Elements of Mathematics) (Springer, Berlin, 2002)CrossRefMATHGoogle Scholar
  7. 20.
    J.W.S. Cassels, An Introduction to the Geometry of Numbers (Springer, Berlin, 1997)MATHGoogle Scholar
  8. 21.
    H. Cohen, A Course in Computational Algebraic Number Theory (Springer, New York, 1996)Google Scholar
  9. 23.
    J. Conway, N. Sloane, Voronoi regions of lattices, second moments of polytopes, and quantization. IEEE Trans. Inf. Theory 28(2), 211–226 (1982)MathSciNetCrossRefMATHGoogle Scholar
  10. 24.
    J.H. Conway, N.J.A. Sloane, Laminated lattices. Ann. Math. 116(3), 593–620 (1982)MathSciNetCrossRefMATHGoogle Scholar
  11. 25.
    J.H. Conway, N.J.A. Sloane, On the voronoi regions of certain lattices. SIAM J. Algebr. Discret. Methods 5(3), 294–305, (1984)MathSciNetCrossRefMATHGoogle Scholar
  12. 26.
    J.H. Conway, N.J.A. Sloane, Sphere-Packings, Lattices, and Groups (Springer, New York, 1998)MATHGoogle Scholar
  13. 72.
    D. Micciancio, S. Goldwasser, Complexity of Lattice Problems. The Kluwer International Series in Engineering and Computer Science, vol. 671 (Kluwer Academic Publishers, Boston, MA, 2002). A cryptographic perspectiveGoogle Scholar
  14. 73.
    D. Micciancio, O. Regev, Lattice-Based Cryptography. Post-Quantum Cryptography (Springer, Berlin, 2009)CrossRefMATHGoogle Scholar
  15. 77.
    G. Nebe, E.M. Rains, N.J.A. Sloane, A simple construction for the barnes-wall lattices, in Codes, Graphs, and Systems (Springer, Berlin, 2002), pp. 333–342MATHGoogle Scholar
  16. 86.
    C.A. Rogers, Packing and Covering (Cambridge University Press, Cambridge, 1964)MATHGoogle Scholar
  17. 90.
    A. Schurmann, F. Vallentin, Computational approaches to lattice packing and covering problems. Discret. Comput. Geom. 35(1), 73–116 (2006)MathSciNetCrossRefMATHGoogle Scholar
  18. 98.
    N.J.A. Sloane, The sphere packing problem. Doc. Math. Extra Volume ICM, 387–396 (1998)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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