BIBD-s for (13, 5, 5), (16, 6, 5) and (21, 6, 4) Possessing Possibly an Automorphism of Order 3
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.
[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, http://www.emba.uvm.edu/ 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.
[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, http://www.emba.uvm.edu/ 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.
@article{"International Journal of Engineering, Mathematical and Physical Sciences:58060", author = "Ivica Martinjak and Mario-Osvin Pavcevic", title = "BIBD-s for (13, 5, 5), (16, 6, 5) and (21, 6, 4) Possessing Possibly an Automorphism of Order 3", abstract = "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 = "BIBD, incidence matrix, automorphism group, tactical decomposition, deterministic algorithm.", volume = "3", number = "10", pages = "833-4", }