Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 31108
BIBD-s for (13, 5, 5), (16, 6, 5) and (21, 6, 4) Possessing Possibly an Automorphism of Order 3

Authors: Ivica Martinjak, Mario-Osvin Pavcevic


When trying to enumerate all BIBD-s for given parameters, their natural solution space appears to be huge and grows extremely with the number of points of the design. Therefore, constructive enumerations are often carried out by assuming additional constraints on design-s structure, automorphisms being mostly used ones. It remains a hard task to construct designs with trivial automorphism group – those with no additional symmetry – although it is believed that most of the BIBD-s belong to that case. In this paper, very many new designs with parameters 2-(13, 5, 5), 2-(16, 6, 5) and 2-(21, 6, 4) are constructed, assuming an action of an automorphism of order 3. Even more, it was possible to construct millions of such designs with no non-trivial automorphisms.

Keywords: automorphism group, BIBD, incidence matrix, tactical decomposition, deterministic algorithm

Digital Object Identifier (DOI):

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1050


[1] V. Cepulic, On symmetric block designs (40, 13, 4) with automorphisms of order 5, Discrete Math. 128(1-3) (1994), 45-60.
[2] C. J. Colbourn, J. H. Dinitz, The CRC Handbook of Combinatorial Designs New Results, dinitz/newresults.html
[3] Z. Janko and Tran van Trung, Construction of a new symmetric block design for (78,22,6) with the help of tactical decompositions, J. Comb. Theory Ser.A 40 (1985), 451-455.
[4] V. Krˇcadinac, Konstrukcija i klasifikacija konaˇcnih struktura pomo'cu raˇcunala, doktorska disertacija, Sveuˇciliˇste u Zagrebu, 2004.
[5] E. Lander, Symmetric Designs: An Algebraic Approach, Cambridge University Press, Cambridge, England, 1983.
[6] R. Mathon, A. Rosa, 2-(v, k, λ) Designs of Small Order, in CRC Handbook of Combinatorial Designs Second Edition, C. J. Colbourn and J. H. Dinitz (Editors), CRC Press, Boca Raton, FL, 2007, pp. 25-58.
[7] B. D. McKay, nauty user-s guide (version 1.5), Technical Report TR-CS-90-02, Department of Computer Science, Australian National University,1990.