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Mathematica Slovaca

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Volume 59, Issue 2 (Apr 2009)

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Classification of resolvable balanced incomplete block designs — the unitals on 28 points

Petteri Kaski
  • Department of Computer Science Helsinki Institute for Information Technology HIIT, University of Helsinki, P.O. Box 68, 00014, University of Helsinki, Finland
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/ Patric Östergård
  • Department of Electrical and Communications Engineering, Helsinki University of Technology, P.O. Box 3000, 02015 TKK, Helsinki, Finland
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Published Online: 2009-03-13 | DOI: https://doi.org/10.2478/s12175-009-0113-8

Abstract

Approaches for classifying resolvable balanced incomplete block designs (RBIBDs) are surveyed. The main approaches can roughly be divided into two types: those building up a design parallel class by parallel class and those proceeding point by point. With an algorithm of the latter type — and by refining ideas dating back to 1917 and the doctoral thesis by Pieter Mulder — it is shown that the list of seven known resolutions of 2-(28, 4, 1) designs is complete; these objects are also known as the resolutions of unitals on 28 points.

MSC: Primary 05B05

Keywords: classification; RBIBD; resolution; Steiner system; unital

  • [1] APPA, G.— MAGOS, D.— MOURTOS, I.: An LP-based proof for the non-existence of a pair of orthogonal Latin squares of order 6, Oper. Res. Lett. 32 (2004), 336–344. http://dx.doi.org/10.1016/j.orl.2003.10.010CrossrefGoogle Scholar

  • [2] BETH, T.— JUNGNICKEL, D.— LENZ, H.: Design Theory, Vol. I, II (2nd ed.), Cambridge University Press, Cambridge, 1999. Google Scholar

  • [3] BROUWER, A. E.: Some unitals on 28 points and their embeddings in projective planes of order 9. In: Geometries and Groups (M. Aigner, D. Jungnickel, eds.). Lecture Notes in Math. 893, Springer, Berlin, 1981, pp. 183–188. http://dx.doi.org/10.1007/BFb0091018Google Scholar

  • [4] COLE, F. N.: Kirkman parades, Bull. Amer. Math. Soc. 28 (1922), 435–437. http://dx.doi.org/10.1090/S0002-9904-1922-03599-9CrossrefGoogle Scholar

  • [5] COLE, F. N.— CUMMINGS, L. D.— WHITE, H. S.: The complete enumeration of triad systems in 15 elements, Proc. Natl. Acad. Sci. USA. 3 (1917), 197–199. http://dx.doi.org/10.1073/pnas.3.3.197CrossrefGoogle Scholar

  • [6] DINITZ, J. H.— GARNICK, D. K.— MCKAY, B. D.: There are 526, 915, 620 nonisomorphic one-factorizations of K 12, J. Combin. Des. 2 (1994), 273–285. http://dx.doi.org/10.1002/jcd.3180020406CrossrefGoogle Scholar

  • [7] FARADŽEV, I. A.: Constructive enumeration of combinatorial objects. In: Problèmes Combinatoires et Théorie des Graphes (Université d’Orsay, July 9–13, 1977), CNRS, Paris, 1978, pp. 131–135 Google Scholar

  • [8] FURINO, S.— MIAO, Y.— YIN, J.: Frames and Resolvable Designs. Uses, Constructions, and Existence, CRC Press, Boca Raton, 1996. Google Scholar

  • [9] HALL, M., Jr.: Combinatorial Theory (2nd ed.), Wiley, New York, 1986. Google Scholar

  • [10] KASKI, P.— MORALES, L. B.— ÖSTERGÅRD, P. R. J.— ROSENBLUETH, D. A.— VELARDE, C.: Classification of resolvable 2-(14, 7, 12) and 3-(14, 7, 5) designs, J. Combin. Math. Combin. Comput. 47 (2003), 65–74. Google Scholar

  • [11] KASKI, P.— ÖSTERGÅRD, P. R. J.: There exists no (15, 5, 4) RBIBD, J. Combin. Des. 9 (2001), 357–362. http://dx.doi.org/10.1002/jcd.1016CrossrefGoogle Scholar

  • [12] KASKI, P.— ÖSTERGÅRD, P. R. J.: Miscellaneous classification results for 2-designs, Discrete Math. 280 (2004), 65–75. http://dx.doi.org/10.1016/j.disc.2003.07.002CrossrefGoogle Scholar

  • [13] KASKI, P.— ÖSTERGÅRD, P. R. J.: The Steiner triple systems of order 19, Math. Comp. 73 (2004), 2075–2092. http://dx.doi.org/10.1090/S0025-5718-04-01626-6CrossrefGoogle Scholar

  • [14] KASKI, P.— ÖSTERGÅRD, P. R. J.: One-factorizations of regular graphs of order 12, Electron. J. Combin. 12 (2005) No. 1, #R2, 25pp. Google Scholar

  • [15] KASKI, P.— ÖSTERGÅRD, P. R. J.: Classification Algorithms for Codes and Designs, Springer, Berlin, 2006. Google Scholar

  • [16] KASKI, P.— ÖSTERGÅRD, P. R. J.— POTTONEN, O.: The Steiner quadruple systems of order 16, J. Combin. Theory Ser. A 113 (2006), 1764–1770. http://dx.doi.org/10.1016/j.jcta.2006.03.017CrossrefGoogle Scholar

  • [17] KRČADINAC, V.: Steiner 2-designs S(2, 4, 28) with nontrivial automorphisms, Glas. Mat. Ser. III 37(57) (2002), 259–268. Google Scholar

  • [18] LAM, C. W. H.— THIEL, L.: Backtrack search with isomorph rejection and consistency check, J. Symbolic Comput. 7 (1989), 473–485. http://dx.doi.org/10.1016/S0747-7171(89)80029-XCrossrefGoogle Scholar

  • [19] LAM, C.— TONCHEV, V. D.: Classification of affine resolvable 2-(27, 9, 4) designs, J. Statist. Plann. Inference 56; 86 (1996; 2000), 187–202; 277–278. http://dx.doi.org/10.1016/S0378-3758(96)00018-3CrossrefGoogle Scholar

  • [20] MATHON, R.— LOMAS, D.: A census of 2-(9, 3, 3) designs, Australas. J. Combin. 5 (1992), 145–158. Google Scholar

  • [21] MATHON, R.— ROSA, A.: A census of Mendelsohn triple systems of order nine, Ars Combin. 4 (1977), 309–315. Google Scholar

  • [22] MATHON, R.— ROSA, A.: Some results on the existence and enumeration of BIBD’s. Mathematics Report 125-Dec-1985, Department of Mathematics and Statistics, McMaster University, Hamilton, 1985. Google Scholar

  • [23] MATHON, R.— ROSA, A.: Tables of parameters of BIBDs with r ≤ 41 including existence, enumeration, and resolvability results, Ann. Discrete Math. 26 (1985), 275–307. Google Scholar

  • [24] MATHON, R.— ROSA, A.: Tables of parameters of BIBDs with r ≤ 41 including existence, enumeration and resolvability results: An update, Ars Combin. 30 (1990), 65–96. Google Scholar

  • [25] MATHON, R.— ROSA, A.: 2-(υ, k, λ) designs of small order. In: The CRC Handbook of Combinatorial Designs (C. J. Colbourn, J. H. Dinitz, eds.), CRC Press, Boca Raton, 1996, pp. 3–41. Google Scholar

  • [26] MATHON, R.— ROSA, A.: 2-(υ, k, λ) designs of small order. In: Handbook of Combinatorial Designs (C. J. Colbourn, J. H. Dinitz, eds.; 2nd ed.), Chapman & Hall/CRC Press, Boca Raton, 2007, pp. 25–58. Google Scholar

  • [27] MCKAY, B. D.: nauty User’s guide (version 1.5). Technical Report TR-CS-90-02, Computer Science Department, Australian National University, Canberra, 1990. Google Scholar

  • [28] MCKAY, B. D.: autoson —A distributed batch system for UNIX workstation networks (version 1.3). Technical Report TR-CS-96-03, Computer Science Department, Australian National University, Canberra, 1996. Google Scholar

  • [29] MCKAY, B. D.: Isomorph-free exhaustive generation, J. Algorithms 26 (1998), 306–324. http://dx.doi.org/10.1006/jagm.1997.0898CrossrefGoogle Scholar

  • [30] MORALES, L. B.— VELARDE, C.: A complete classification of (12, 4, 3)-RBIBDs, J. Combin. Des. 9 (2001), 385–400. http://dx.doi.org/10.1002/jcd.1019CrossrefGoogle Scholar

  • [31] MORALES, L. B.— VELARDE, C.: Enumeration of resolvable 2-(10, 5, 16) and 3-(10, 5, 6) designs, J. Combin. Des. 13 (2005), 108–119. http://dx.doi.org/10.1002/jcd.20032CrossrefGoogle Scholar

  • [32] MORGAN, E. J.: Some small quasi-multiple designs, Ars Combin. 3 (1977), 233–250. Google Scholar

  • [33] MULDER, P.: Kirkman-Systemen. PhD Thesis, Rijksuniversiteit Groningen, 1917. Google Scholar

  • [34] NISKANEN, S.— ÖSTERGÅRD, P. R. J.: Cliquer user’s guide, version 1.0. Technical Report T48, Communications Laboratory, Helsinki University of Technology, Espoo, 2003. Google Scholar

  • [35] ÖSTERGÅRD, P. R. J.: Enumeration of 2-(12, 3, 2) designs, Australas. J. Combin. 22 (2000), 227–231. Web of ScienceGoogle Scholar

  • [36] ÖSTERGÅRD, P. R. J.— KASKI, P.: Enumeration of 2-(9, 3, λ) designs and their resolutions, Des. Codes Cryptogr. 27 (2002), 131–137. http://dx.doi.org/10.1023/A:1016558720904CrossrefGoogle Scholar

  • [37] PENTTILA, T.— ROYLE, G. F.: Sets of type (m, n) in the affine and projective planes of order nine, Des. Codes Cryptogr. 6 (1995), 229–245. http://dx.doi.org/10.1007/BF01388477CrossrefGoogle Scholar

  • [38] READ, R. C.: Every one a winner; or, How to avoid isomorphism search when cataloguing combinatorial configurations, Ann. Discrete Math. 2 (1978), 107–120. http://dx.doi.org/10.1016/S0167-5060(08)70325-XCrossrefGoogle Scholar

  • [39] SEMAKOV, N. V.— ZINOV’EV, V. A.: Equidistant q-ary codes with maximal distance and resolvable balanced incomplete block designs, Problemy Peredachi Informatsii 4 (1968), No. 2, 3–10 (Russian). [English translation: Probl. Inf. Transm. 4 (1968), No. 2, 1–7]. Google Scholar

  • [40] WHITE, H. S.— COLE, F. N.— CUMMINGS, L. D.: Complete classification of triad systems on fifteen elements, Memoirs Natl. Acad. Sci. USA. 27 (1919) No. 2, 1–89. Google Scholar

About the article

Published Online: 2009-03-13

Published in Print: 2009-04-01


Citation Information: Mathematica Slovaca, ISSN (Online) 1337-2211, ISSN (Print) 0139-9918, DOI: https://doi.org/10.2478/s12175-009-0113-8.

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© 2009 Mathematical Institute, Slovak Academy of Sciences. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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