Abstract
The aim of this paper is to present a recursive construction of simple t-designs for arbitrary t. The construction is of purely combinatorial nature and it requires finding solutions for the indices of the ingredient designs that satisfy a certain set of equalities. We give a small number of examples to illustrate the construction, whereby we have found a large number of new t-designs, which were previously unknown. This indicates that the method is useful and powerful.
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References
Ajoodani-Namini S.: Extending large sets of \(t\)-designs. J. Combin. Theory A 76, 139–144 (1996).
Beth T., Jungnickel D., Lenz H.: Design Theory, 2nd edn. Cambridge University Press, Cambridge (1999).
Betten A., Kerber A., Kohnert A., Laue R., Wassermann A.: The discovery of simple 7-designs with automorphism group \(P\Gamma L(2,32)\). In: Cohen G., Giusti M., Mora T. (eds.) Applied Algebra, Algebraic Algorithms and Error-Correcting Codes, pp. 131–145. Springer, New York (1995).
Betten A., Kerber A., Laue R., Wassermann A.: Simple 8-designs with small parameters. Des. Codes Cryptogr. 15, 5–27 (1998).
Betten A., Laue R., Wassermann A.: A Steiner 5-design on 36 points. Des. Codes Cryptogr. 17, 181–186 (1999).
Bierbrauer J.: A family of 4-designs with block size 9. Discret. Math. 138, 113–117 (1995).
Bierbrauer J.: A family of 4-designs. Graphs Comb. 11, 209–212 (1995).
Colbourn C. J., Dinitz J. H., (eds.): Handbook of Combinatorial Designs, 2nd edn. CRC Press, Boca Raton (2007).
Denniston R.H.F.: Some new 5-designs. Bull. Lond. Math. Soc. 8, 263–267 (1976).
Driessen L.H.M.E.: \(t\)-Designs, \(t\ge 3\). Technical Report, Department of Mathematics, Technische Hogeschool Eindhoven, The Netherlands (1978).
Jimbo M., Kunihara Y., Laue R., Sawa M.: Unifying some infinite families of combinatorial 3-designs. J. Combin. Theory A 118, 1072–1085 (2011).
Khosrovshahi G.B., Ajoodani-Namini S.: Combining \(t\)-designs. J. Combin. Theory Ser. A 58, 26–34 (1991).
Kramer E.S., Mesner D.M.: \(t\)-Designs on hypergraphs. Discret. Math. 15, 263–296 (1976).
Kramer E.S., Magliveras S.S., O’Brien E.A.: Some new large sets of t-designs. Australas. J. Combin. 7, 189–193 (1993).
Kreher D.L.: An infinite family of (simple) 6-designs. J. Combin. Des. 1, 277–280 (1993).
Magliveras S.S., Leavitt D.M.: Simple 6-(33,8,36)-Designs from \(P\Gamma L_2(32)\), Computational Group Theory, pp. 337–352. Academic Press, New York (1984).
Magliveras S.S., Plambeck T.E.: New infinite families of simple 5-designs. J. Combin. Theory A 44, 1–5 (1987).
Sebille M.: There exists a simple non-trivial \(t\)-design with an arbitrarily large automorphism group for every \(t\). Des. Codes Cryptogr. 22, 203–206 (2001).
Teirlinck L.: Non-trivial \(t\)-designs without repeated blocks exist for all \(t\). Discret. Math. 65, 301–311 (1987).
Teirlinck L.: Locally trivial \(t-\)designs and \(t\)-designs without repeated blocks. Discret. Math. 77, 345–356 (1989).
Tuan N.D.: Simple non-trivial designs with an arbitrary automorphism group. J. Combin. Theory A 100, 403–408 (2002).
van Trung T.: The existence of an infinite family of simple 5-designs. Math. Z. 187, 285–287 (1984).
van Trung T.: On the construction of \(t\)-designs and the existence of some new infinite families of simple 5-designs. Arch. Math. 47, 187–192 (1986).
van Trung T.: Recursive constructions for 3-designs and resolvable 3-designs. J. Stat. Plan. Infer. 95, 341–358 (2001).
Wu Q.: A note on extending \(t\)-designs. Australas. J. Comb. 4, 229–235 (1991).
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Communicated by J. D. Key.
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van Trung, T. Simple t-designs: a recursive construction for arbitrary t . Des. Codes Cryptogr. 83, 493–502 (2017). https://doi.org/10.1007/s10623-016-0238-z
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DOI: https://doi.org/10.1007/s10623-016-0238-z