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Covering radius for codes obtained from T(m) triangular graphs

  • J. M. Basart
  • J. Rifà
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 356)

Abstract

Triangular graphs are a special case of the well-known strongly regular graphs.

Taking any spanning tree in a T(m) triangular graph -m≥4- we get a fundamental circuit matrix for it. Using this matrix as a generator matrix we can obtain a single-error-correcting linear code C(T(m)) with parameters:

n=(m(m−1) (m−2))/2, k=(m(m−1) (m−3)+2)/2 and d=3.

Using the fact that each codeword in C(T(m)) is formed by a combination of simple circuits in T(m), we give a characterization of its codewords which allow us to show that:
  1. (i)

    whatever the value of m be, if we take a hamiltonian path as a spanning tree in T(m), the obtained code C(T(m)) has covering radius σ equal to [m(m−1)/4] and

     
  2. (ii)

    fixed m all the C(T(m)) codes are equivalent independently of the chosen spanning tree,

     

so, it is finally proved that given T(m) and any spanning tree in it, C(T(m)) has σ=[m(m−1)/4].

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References

  1. [1]
    Basart J.M. and Huguet L. "From T(m) triangular graphs to single-error-correcting codes" Proceedings of the conference AAECC-IV (Karlsruhe 1986) Lecture Notes in Computer Science vol. 307 Springer-Verlag 1988.Google Scholar
  2. [2]
    Rose R.C. "Strongly regular graphs, partial geometries and partially balanced designs" Pacific J. Math. 13, 1963.Google Scholar
  3. [3]
    Cameron P.J. and Van Lint J.H. "Graph theory, coding theory and block designs" Cambridge University Press, 1975.Google Scholar
  4. [4]
    Christofides N. "Graph theory, an algorithmic approach" Academic Press 1975.Google Scholar
  5. [5]
    Goethals J.M. and Seidel J.J. "Strongly regular graphs from combinatorial designs" Canadian J. Math. 3, 1970.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • J. M. Basart
    • 1
  • J. Rifà
    • 1
  1. 1.Departament d'Informàtica, Facultat de CiènciesUniversitat Autònoma de BarcelonaBellaterra Catalonia

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