Advertisement

Designs, Codes and Cryptography

, Volume 17, Issue 1–3, pp 187–209 | Cite as

Binary Codes of Strongly Regular Graphs

  • Willem H. Haemers
  • René Peeters
  • Jeroen M. van Rijckevorsel
Article

Abstract

For strongly regular graphs ith adjacency matrix A, we look at the binary codes generated by A and A + I. We determine these codes for some families of graphs, e pay attention to the relation beteen the codes of switching equivalent graphs and, ith the exception of two parameter sets, we generate by computer the codes of all knon strongly regular graphs on fewer than 45 vertices.

binary codes strongly regular graphs regular two-graphs 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. L. Arlazarov, A. A. Lehman, M. Z. Rosenfeld, Computer-Aided Construction and Analysis of Graphs with 25, 26 and 29 Vertices, Institute of Control Problems, Moscow (1975).Google Scholar
  2. 2.
    E. F. Assmus Jr. and A. A. Drisko, Binary codes of odd-order nets, Designs, Codes and Cryptography, Vol. 17 (1999) pp. 15-36.Google Scholar
  3. 3.
    E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge tracts in mathematics, 103, Cambridge University Press (1992).Google Scholar
  4. 4.
    E. F. Assmus Jr. and J. D. Key, Designs and Codes: An Update, Designs, Codes and Cryptography, Vol. 9 (1996) pp. 7-27.Google Scholar
  5. 5.
    A. E. Brouwer and C. A. van Eijl, On the p-Rank of the Adjacency Matrices of Strongly Regular Graphs, J. Algebraic Combin., Vol. 1 (1992) pp. 329-346.Google Scholar
  6. 6.
    A. E. Brouwer and H. A. Wilbrink, Block Designs, Chap. 8, Handbook of Incidence Geometry, Buildings and Foundations, (F. Buekenhout, ed.), North-Holland (1995) pp. 349-382.Google Scholar
  7. 7.
    A. E. Brouwer, H. A. Wilbrink and W. H. Haemers, Some 2-ranks, Discrete Math., Vol. 106/107 (1992) pp. 83-92.Google Scholar
  8. 8.
    F. C. Bussemaker and J. J. Seidel, Symmetric Hadamard matrices of order 36, Report 70-WSK-02, Technical University Eindhoven (1970).Google Scholar
  9. 9.
    F. C. Bussemaker, R. Mathon and J. J. Seidel, Tables of two-graphs, Report 79-WSK-05, Technical University Eindhoven (1979).Google Scholar
  10. 10.
    P. J. Cameron and J. H. van Lint, Designs, Graphs, Codes and Their Links, Cambridge University Press (1991).Google Scholar
  11. 11.
    J. Doyen, X. Hubaut and M. Vandensavel, Ranks of Incidence Matrices of Steiner Triple Systems, Math Z., Vol. 163, Springer-Verlag (1978) pp. 251-259.Google Scholar
  12. 12.
    W. H. Haemers, C. Parker, V. Pless and V. D. Tonchev, A Design and a Code Invariant under the Simple Group Co3, J. Combin Theory Ser A, Vol. 62 (1993) pp. 225-233.Google Scholar
  13. 13.
    J. S. Leon, Backtrack partition programs, ftp://math.uic.edu/pub/leon/partn.Google Scholar
  14. 14.
    J. Mac Williams and N. Sloane, The Theory of Error-Correcting Codes, North-Holland, Mathematical Library (1977).Google Scholar
  15. 15.
    G. E. Moorhouse, Bruck Nets, Codes, and Characters of Loops, Designs, Codes and Cryptography, Vol. 1 (1991) pp. 7-29.Google Scholar
  16. 16.
    A. J. L. Paulus, Conference Matrices and Graphs of Order 26, T.H.-Report 73-WSK-06 (1973).Google Scholar
  17. 17.
    René Peeters, Ranks and Structure of Graphs, dissertation, Tilburg University (1995).Google Scholar
  18. 18.
    E. Spence, (40,13,4)-Designs derived from strongly regular graphs, Advances in Finite Geometries and Designs, Proc. of the Third Isle of Thorns Conf. 1990 (J. W. P. Hirschfeld, D. R. Highes and J. A. Thas, eds.), Oxford Science (1991) pp. 359-368.Google Scholar
  19. 19.
    E. Spence, Regular Two-graphs on 36 Vertices, Linear Algebra Appl., Vol. 226–228 (1995) pp. 459-497.Google Scholar
  20. 20.
    E. Spence, personal communication.Google Scholar
  21. 21.
    V. D. Tonchev, Codes, The CRC Handbook of Combinatorial Designs (C. J. Colbourn and J. H. Dinitz, eds.) (1996) pp. 517-543.Google Scholar
  22. 22.
    V. D. Tonchev, Binary codes derived from the Hoffman-Singleton and Higman-Sims graphs, IEEE Trans. Inform. Theory, Vol. 43 (1997) pp. 1021-1025.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • Willem H. Haemers
    • 1
  • René Peeters
    • 1
  • Jeroen M. van Rijckevorsel
    • 1
  1. 1.Department of EconometricsTilburg UniversityTilburgThe Netherlands

Personalised recommendations