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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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).
E. F. Assmus Jr. and A. A. Drisko, Binary codes of odd-order nets, Designs, Codes and Cryptography, Vol. 17 (1999) pp. 15-36.
E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge tracts in mathematics, 103, Cambridge University Press (1992).
E. F. Assmus Jr. and J. D. Key, Designs and Codes: An Update, Designs, Codes and Cryptography, Vol. 9 (1996) pp. 7-27.
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.
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.
A. E. Brouwer, H. A. Wilbrink and W. H. Haemers, Some 2-ranks, Discrete Math., Vol. 106/107 (1992) pp. 83-92.
F. C. Bussemaker and J. J. Seidel, Symmetric Hadamard matrices of order 36, Report 70-WSK-02, Technical University Eindhoven (1970).
F. C. Bussemaker, R. Mathon and J. J. Seidel, Tables of two-graphs, Report 79-WSK-05, Technical University Eindhoven (1979).
P. J. Cameron and J. H. van Lint, Designs, Graphs, Codes and Their Links, Cambridge University Press (1991).
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.
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.
J. S. Leon, Backtrack partition programs, ftp://math.uic.edu/pub/leon/partn.
J. Mac Williams and N. Sloane, The Theory of Error-Correcting Codes, North-Holland, Mathematical Library (1977).
G. E. Moorhouse, Bruck Nets, Codes, and Characters of Loops, Designs, Codes and Cryptography, Vol. 1 (1991) pp. 7-29.
A. J. L. Paulus, Conference Matrices and Graphs of Order 26, T.H.-Report 73-WSK-06 (1973).
René Peeters, Ranks and Structure of Graphs, dissertation, Tilburg University (1995).
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.
E. Spence, Regular Two-graphs on 36 Vertices, Linear Algebra Appl., Vol. 226–228 (1995) pp. 459-497.
E. Spence, personal communication.
V. D. Tonchev, Codes, The CRC Handbook of Combinatorial Designs (C. J. Colbourn and J. H. Dinitz, eds.) (1996) pp. 517-543.
V. D. Tonchev, Binary codes derived from the Hoffman-Singleton and Higman-Sims graphs, IEEE Trans. Inform. Theory, Vol. 43 (1997) pp. 1021-1025.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Haemers, W.H., Peeters, R. & Rijckevorsel, J.M.v. Binary Codes of Strongly Regular Graphs. Designs, Codes and Cryptography 17, 187–209 (1999). https://doi.org/10.1023/A:1026479210284
Issue Date:
DOI: https://doi.org/10.1023/A:1026479210284