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Lattices from 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

A natural way of constructing lattices is from error-correcting codes, using the so-called Construction A. It associates a lattice in \(\mathbb {R}^{n}\) to a linear code in \(\mathbb {Z}_{q}^{n}\) (the set \(\mathbb {Z}_{q}\) of integers modulo q will be introduced next). Such lattices are also called q-ary lattices (or modulo-q lattices) and have several applications in information theory and cryptography.

References

  1. 2.
    E. Agrell, T. Eriksson, A. Vardy, K. Zeger, Closest point search in lattices. IEEE Trans. Inf. Theory 48(8), 2201–2214 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 19.
    A. Campello, G.C. Jorge, J.E. Strapasson, S.I.R. Costa, Perfect codes in the l p metric. Eur. J. Comb. 53(C), 72–85 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 21.
    H. Cohen, A Course in Computational Algebraic Number Theory (Springer, New York, 1996)Google Scholar
  4. 26.
    J.H. Conway, N.J.A. Sloane, Sphere-Packings, Lattices, and Groups (Springer, New York, 1998)zbMATHGoogle Scholar
  5. 32.
    S.I.R. Costa, A. Campello, G.C. Jorge, J.E. Strapasson, C. Qureshi, Codes and lattices in the lp metric, in 2014 Information Theory and Applications Workshop (ITA) (2014), pp. 1–4Google Scholar
  6. 33.
    W. Ebeling, Lattices and Codes (Springer, Berlin, 2013)CrossRefzbMATHGoogle Scholar
  7. 38.
    T. Etzion, A. Vardy, E. Yaakobi, Coding for the Lee and Manhattan metrics with weighing matrices. IEEE Trans. Inf. Theory 59(10), 6712–6723 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 39.
    C. Feng, D. Silva, F.R. Kschischang, An algebraic approach to physical-layer network coding. IEEE Trans. Inf. Theory 59(11), 7576–7596 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 45.
    S. Golomb, A general formulation of error matrices (corresp.). IEEE Trans. Inf. Theory 15(3), 425–426 (1969)Google Scholar
  10. 46.
    S.W. Golomb, L.R. Welch, Perfect codes in the Lee metric and the packing of polyominoes. SIAM J. Appl. Math. 18, 302–317 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 49.
    B. Hassibi, H. Vikalo, On the sphere-decoding algorithm I. Expected complexity. IEEE Trans. Signal Process. 53(8), 2806–2818 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 50.
    P. Horak, O. Grosek, A new approach towards the Golomb–Welch conjecture. Eur. J. Comb. 38, 12–22 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 53.
    W.C. Huffman, V. Pless, Fundamentals of Error-Correcting Codes (Cambridge University Press, Cambridge, 2010)zbMATHGoogle Scholar
  14. 56.
    G.C. Jorge, Reticulados q-ários e algébricos (in Portuguese), PhD thesis, University of Campinas, 2012Google Scholar
  15. 57.
    G.C. Jorge, A. Campello, S.I.R. Costa, q-ary lattices in the l p norm and a generalization of the Lee metric, in Proceedings of The International Workshop on Coding and Cryptography (WCC) (2013), pp. 15–19Google Scholar
  16. 59.
    W. Kositwattanarerk, S.S. Ong, F. Oggier, Construction a of lattices over number fields and block fading (wiretap) coding. IEEE Trans. Inf. Theory 61(5), 2273–2282 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 60.
    C.C. Lavor, M.M.S. Alves, R.M. Siqueira, S.I.R. Costa, Uma introdução à teoria de códigos. Notas em Matemática Aplicada, vol. 21, (Sociedade Brasileira de Matemática Aplicada e Computacional (SBMAC), 2006)Google Scholar
  18. 61.
    C. Lee, Some properties of nonbinary error-correcting codes. IRE Trans. Inf. Theory 4(2), 77–82 (1958)MathSciNetCrossRefGoogle Scholar
  19. 62.
    S. Leung-Yan-Cheong, M. Hellman, Concerning a bound on undetected error probability (corresp.). IEEE Trans. Inf. Theory 22(2), 235–237 (1976)Google Scholar
  20. 63.
    C. Ling, L. Luzzi, J.-C. Belfiore, Lattice codes with strong secrecy over the mod-λ gaussian channel, in IEEE International Symposium on Information Theory (2012), pp. 2306–2310Google Scholar
  21. 65.
    J. Lu, J. Harshan, F.E. Oggier, Performance of lattice coset codes on a USRP testbed. CoRR, abs/1607.07163 (2016)Google Scholar
  22. 66.
    F.J. MacWilliams, N.J.A. Sloane, The Theory of Error-Correcting Codes (North-Holland, Amsterdam, 1955)zbMATHGoogle Scholar
  23. 73.
    D. Micciancio, O. Regev, Lattice-Based Cryptography. Post-Quantum Cryptography (Springer, Berlin, 2009)CrossRefzbMATHGoogle Scholar
  24. 80.
    F. Oggier, P. Solé, J.C. Belfiore, Lattice codes for the wiretap gaussian channel: construction and analysis. IEEE Trans. Inf. Theory 62(10), 5690–5708 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 81.
    L.H. Ozarow, A.D. Wyner, Wire-tap channel II. AT&T Bell Lab. Tech. J. 63(10), 2135–2157 (1984)CrossRefzbMATHGoogle Scholar
  26. 82.
    C. Peikert, Limits on the hardness of lattice problems in l p norms, in IEEE 27th Conference on Computational Complexity (2007), pp. 333–346Google Scholar
  27. 85.
    C. Qureshi, S.I.R Costa, On perfect q-ary codes in the maximum metric, in 2014 Information Theory and Applications Workshop (ITA) (2016), pp. 1–4Google Scholar
  28. 88.
    R.M. Roth, P.H. Siegel, Lee-metric BCH codes and their application to constrained and partial-response channels. IEEE Trans. Inf. Theory 40(4), 1083–1096 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 89.
    J.A. Rush, N.J.A. Sloane, An improvement to the Minkowski-Hlawka bound for packing superballs. Mathematika 34, 8–18 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 94.
    M.Z. Shieh, S.C. Tsai, Decoding frequency permutation arrays under chebyshev distance. IEEE Trans. Inf. Theory 56(11), 5730–5737 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 98.
    N.J.A. Sloane, The sphere packing problem. Doc. Math. Extra Volume ICM, 387–396 (1998)Google Scholar
  32. 101.
    Y. Song, N. Devroye, Lattice codes for the gaussian relay channel: decode-and-forward and compress-and-forward. IEEE Trans. Inf. Theory 59(8), 4927–4948 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 109.
    E. Viterbo, J. Boutros, A universal lattice code decoder for fading channels. IEEE Trans. Inf. Theory 45(5), 1639–1662 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 111.
    R. Zamir, Lattices are everywhere, in IEEE Xplore (ed.), Information Theory and Applications Workshop (2009), pp. 392–421Google 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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