Computational Methods in Design Theory

  • Rudolf Mathon


Enumeration theory, which aims to count the number of distinct (non-equivalent) elements in a given class of combinatorial objects, constitutes a significant area in combinatorial analysis. The object of constructive enumeration consists of creating a complete list of configurations with given properties [5], [8]. There are several reasons which stimulate research in constructive enumeration. Classical methods are not applicable to many interesting classes of objects such as strongly regular graphs, combinatorial designs, error correcting codes, etc. At present, the only available way to count them is by using algorithmic techniques for fixed values of parameters. Lists of objects are important for generating and testing various hypotheses about invariants, characterization, etc. Moreover, examples of designs with given properties are needed in many areas of applied combinatorics such as coding and experiment planning theories, network reliability and cryptography. Algorithms for constructive enumeration frequently require searching in high dimensional spaces and employ sophisticated techniques to identify partial (final) solutions. Such methods may be of independent interest in artificial intelligence, computer vision, neural networks and combinatorial optimization.


Local Search Automorphism Group Regular Graph Discrete Math Design Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    T. Beth, D. Jungnickel, H. Lenz, Design Theory ( Bibliographishes Institut, Manheim-Wien-Zurich, 1985 ).zbMATHGoogle Scholar
  2. [2]
    M.J. Colbourn, Algorithmic aspects of combinatorial designs: A survey, Ann. Discrete Math. 26 (1985), 67–136.Google Scholar
  3. [3]
    M.J. Colbourn, R.A. Mathon, On cyclic Steiner 2-designs, Ann. Discrete Math. 7 (1980), 215–253.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    F.N.Cole, L.D. Cummings, H.S. White, Complete classification of the triad systems on fifteen elements, Memoirs of the National Academy of Sciences U.S.A. 14, Second memoir (1919), 1–89.Google Scholar
  5. [6]
    K. Engel, H.-D.O.F. Gronau, On 2-(6,3,1) designs, Rostock. Math. Kolloq. 34 (1988), 37–48.MathSciNetzbMATHGoogle Scholar
  6. [7]
    P.B. Gibbons, Computing techniques for the construction and analysis of block designs, PhD Thesis, University of Toronto (1976).Google Scholar
  7. [8]
    A.V. Ivanov, Constructive enumeration of incidence systems, Ann. Discrete Math. 26 (1985), 227–246.Google Scholar
  8. [9]
    D.S. Johnson, C.A. Aragon, L.A. McGeoch, C. Schevon, Optimization by simulated annealing: An experimental evaluation. Part I, Graph partitioning Operations Research 37 (1989), 865–892.CrossRefzbMATHGoogle Scholar
  9. [10]
    D.S. Johnson, C.A. Aragon, L.A. McGeoch, C. Schevon, Optimization by simulated annealing: An experimental evaluation. Part III, Graph coloring and number partitioning, Operations Research 39 (1991), 378–406.CrossRefzbMATHGoogle Scholar
  10. [11]
    E.S. Kramer, D.M. Mesner, t-designs on hypergraphs, Discrete Math. 15 (1976), 263–296.zbMATHGoogle Scholar
  11. [12]
    E.S. Kramer, S.S. Magliveras, R. Mathon, The Steiner systems S(2,4,25) with nontrivial automorphism group, Discrete Math. 77 (1989), 137–157.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [13]
    E.S. Kramer, D.W. Leavitt, S.S. Magliveras, Construction procedures for t-designs and the existence of new simple 6-designs, Ann. Discrete Math. 26 (1985), 247–274.MathSciNetGoogle Scholar
  13. [14]
    D.L. Kreher, S.P. Radziszowski, Finding simple t-designs by using basis reduction, Congressus Numerantium 55 (1986), 235–244.MathSciNetGoogle Scholar
  14. [15]
    D.L. Kreher, S.P. Radziszowski, The existence of simple 6- (14,7,4) designs, J. Combinatorial Theory 43A (1986), 237–243.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [16]
    S.S. Magliveras, D.W. Leavitt, Simple 6-(33,8,36) Designs from PGL2(32), Proceedings of the Durham Computational Group Theory Symposium (Academic Press, 1983 ), 337–352.Google Scholar
  16. [17]
    R. Mathon, Symmetric (31,10,3) designs with nontrivial automorphism group, Ars Combinatoria 25 (1988), 171–183.MathSciNetzbMATHGoogle Scholar
  17. [18]
    R. Mathon, Computational methods in design theory, Surveys in Combinatorics, 1991 (Guildford, 1991 ) 26 (1985), 101–117.MathSciNetGoogle Scholar
  18. [19]
    R. Mathon, D. Lomas, A census of 2-(9,3,3) designs, Austral. J. Combin. 5 (1992), 145–158.MathSciNetzbMATHGoogle Scholar
  19. [20]
    R. Mathon, A. Rosa, Tables of parameters of BIBD’s with r 41 including existence, enumeration, and resolvability results, Ann. Discrete Math. 26 (1985), 275–308.MathSciNetGoogle Scholar
  20. [21]
    D.R. Stinson, Hill-climbing algorithms for the construction of combinatorial designs, Ann. Discrete Math. 26 (1985), 321–334.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1996

Authors and Affiliations

  • Rudolf Mathon
    • 1
  1. 1.Department of Computer ScienceUniversity of TorontoTorontoCanada

Personalised recommendations