Advertisement

Error Correcting Codes from Quasi-Hadamard Matrices

  • V. Álvarez
  • J. A. Armario
  • M. D. Frau
  • E. Martin
  • A. Osuna
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4547)

Abstract

Levenshtein described in [5] a method for constructing error correcting codes which meet the Plotkin bounds, provided suitable Hadamard matrices exist. Uncertainty about the existence of Hadamard matrices on all orders multiple of 4 is a source of difficulties for the practical application of this method. Here we extend the method to the case of quasi-Hadamard matrices. Since efficient algorithms for constructing quasi-Hadamard matrices are potentially available from the literature (e.g. [7]), good error correcting codes may be constructed in practise. We illustrate the method with some examples.

Keywords

Error correcting code Hadamard matrix Hadamard code 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Cameron, P.J.: Combinatorics: topics, techniques, algorithms. Cambridge University Press, Cambridge (1994)zbMATHGoogle Scholar
  2. 2.
    Feige, U., Goldwasser, S., Safra, S., Lovász, L., Szegedy, M.: Approximating clique is almost NP-complete. In: FOCS. Proceedings 32nd Annual Symposium on the Foundations of Computer Science, pp. 2–12 (1991)Google Scholar
  3. 3.
    Hastad, J.: Clique is hard to approximate within n 1 − ε. In: FOCS. Proceedings 37th Annual IEEE Symposium on the Foundations of Computer Science, pp. 627–636. IEEE Computer Society Press, Los Alamitos (1996)Google Scholar
  4. 4.
    Karp, R.M.: Reducibility among combinatorial problems. Complexity of Computer Computations, pp. 85–103 (1972)Google Scholar
  5. 5.
    Levenshtein, V.I.: Application of the Hadamard matrices to a problem in coding. Problems of Cybernetics 5, 166–184 (1964)Google Scholar
  6. 6.
    MacWilliams, F.J., Sloane, N.J.A.: The theory of error-correcting codes. North-Holland, Amsterdam (1977)zbMATHGoogle Scholar
  7. 7.
    Marchiori, E.: Genetic, Iterated and Multistart Local Search for the Maximum Clique Problem. In: Cagnoni, S., Gottlieb, J., Hart, E., Middendorf, M., Raidl, G.R. (eds.) EvoIASP 2002, EvoWorkshops 2002, EvoSTIM 2002, EvoCOP 2002, and EvoPlan 2002. LNCS, vol. 2279, pp. 112–121. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. 8.
    Noboru, I.: Hadamard Graphs I. Graphs Combin. 1 1, 57–64 (1985)zbMATHCrossRefGoogle Scholar
  9. 9.
    Noboru, I.: Hadamard Graphs II. Graphs Combin. 1 4, 331–337 (1985)Google Scholar
  10. 10.
    Plotkin, M.: Binary codes with specified minimum distances. IEEE Trans. Information Theory 6, 445–450 (1960)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • V. Álvarez
    • 1
  • J. A. Armario
    • 1
  • M. D. Frau
    • 1
  • E. Martin
    • 1
  • A. Osuna
    • 1
  1. 1.Dpto. Matemática Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes s/n 41012 SevillaSpain

Personalised recommendations