Advertisement

Links Between Discriminating and Identifying Codes in the Binary Hamming Space

  • Irène Charon
  • Gérard Cohen
  • Olivier Hudry
  • Antoine Lobstein
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4851)

Abstract

Let F n be the binary n-cube, or binary Hamming space of dimension n, endowed with the Hamming distance, and \({\cal E}^n\) (respectively, \({\cal O}^n\)) the set of vectors with even (respectively, odd) weight. For r ≥ 1 and x ∈ F n , we denote by B r (x) the ball of radius r and centre x. A code C ⊆ F n is said to be r-identifying if the sets B r (x) ∩ C, x ∈ F n , are all nonempty and distinct. A code \(C\subseteq {\cal E}^n\) is said to be r-discriminating if the sets B r (x) ∩ C, \(x\in {\cal O}^n\), are all nonempty and distinct. We show that the two definitions, which were given for general graphs,are equivalent in the case of the Hamming space, in the following sense: for any odd r, there is a bijection between the set of r-identifying codes in F n and the set of r-discriminating codes in F n + 1.

Keywords

Graph Theory Coding Theory Discriminating Codes Identifying Codes Hamming Space Hypercub 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Blass, U., Honkala, I., Litsyn, S.: On The Size of Identifying Codes. In: Fossorier, M.P.C., Imai, H., Lin, S., Poli, A. (eds.) AAECC-13. LNCS, vol. 1719, pp. 142–147. Springer, Heidelberg (1999)Google Scholar
  2. 2.
    Blass, U., Honkala, I., Litsyn, S.: On Binary Codes for Identification. J. of Combinatorial Designs 8, 151–156 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Blass, U., Honkala, I., Litsyn, S.: Bounds on Identifying Codes. Discrete Mathematics 241, 119–128 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Charbit, E., Charon, I., Cohen, G., Hudry, O.: Discriminating Codes in Bipartite Graphs. Electronic Notes in Discrete Mathematics 26, 29–35 (2006)CrossRefMathSciNetGoogle Scholar
  5. 5.
    Charon, I., Cohen, G., Hudry, O., Lobstein, A.: Discriminating Codes in (Bipartite) Planar Graphs. European Journal of Combinatorics (to appear)Google Scholar
  6. 6.
    Exoo, G.: Computational Results on Identifying t-codes (preprint, 1999)Google Scholar
  7. 7.
    Exoo, G., Laihonen, T., Ranto, S.: Improved Upper Bounds on Binary Identifying Codes. IEEE Trans. Inform. Theory (to appear)Google Scholar
  8. 8.
    Gimbel, J., Van Gorden, B.D., Nicolescu, M., Umstead, C., Vaiana, N.: Location with Dominating Sets. Congressus Numerantium 151, 129–144 (2001)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Honkala, I., Lobstein, A.: On the Complexity of the Identification Problem in Hamming Spaces. Acta Informatica 38, 839–845 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Karpovsky, M.G., Chakrabarty, K., Levitin, L.B.: On a New Class of Codes for Identifying Vertices in Graphs. IEEE Trans. Inform. Theory 44(2), 599–611 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Ranto, S.: Identifying and Locating-Dominating Codes in Binary Hamming Spaces. Ph. D Thesis, University of Turku (2007)Google Scholar
  12. 12.

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Irène Charon
    • 1
  • Gérard Cohen
    • 1
  • Olivier Hudry
    • 1
  • Antoine Lobstein
    • 1
  1. 1.GET - Télécom Paris & CNRS - LTCI UMR 5141, 46, rue Barrault, 75634 Paris Cedex 13France

Personalised recommendations