Algebraic Coding 1993: Algebraic Coding pp 217-224 | Cite as

Lattices based on linear codes

  • Gregory Poltyrev
Sphere Packings and Lattices
Part of the Lecture Notes in Computer Science book series (LNCS, volume 781)


The Conway and Sloane construction A method is applied to evaluation of a class of lattices. The points of the lattices belong to a scaled set of n-dimensional vectors with integer components and the lattices constructed with the aide of linear over GF(p) codes, p is prime. It is known that the Minkowski-Hlawka bound on the normalized logarithm of packing density is attained asymptotically by such construction of lattices. We show that the capacity of an AWGN channel without restriction can be also attained by these class of lattices.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    G. Poltyrev, ”On Coding without Restrictions for the AWGN Channel”, to be appear in IEEE Trans. Information Theory.Google Scholar
  2. 2.
    J.A. Rush and N.J. Sloane, “An Improvement to the Minkowski Hlawka Bound for Packing Superballs,” Mathematika, vol. 34, pp. 8–18, 1987.Google Scholar
  3. 3.
    J.A. Rush, “A lower bound on packing density,” Inventiones mathematicae, vol. 98, pp. 499–509, 1989.Google Scholar
  4. 4.
    R. Gallager “Low-density parity-check codes”, M.I.T.Press, Cambridge, Massachusets, 1963.Google Scholar
  5. 5.
    H. Herzberg and G. Poltyrev, ”Techniques for Bounding the Probability of Decoding Error in Block Coded Modulation Structures”, to be appear in IEEE Trans. Information Theory.Google Scholar
  6. 6.
    C.E. Shannon, “Probability of error for optimal codes in a Gaussian channel,” Bell Syst. Tech. J., vol. 38, pp. 611–656, May 1959.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • Gregory Poltyrev
    • 1
  1. 1.Department of Electrical Engineering - SystemsTel Aviv UniversityRamat AvivIsrael

Personalised recommendations