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Halvings on small point sets

1999; Wiley; Volume: 7; Issue: 4 Linguagem: Finlandês

10.1002/(sici)1520-6610(1999)7

1520-6610

Reinhard Laue,

Advanced Topology and Set Theory

Journal of Combinatorial DesignsVolume 7, Issue 4 p. 233-241 Halvings on small point sets Reinhard Laue, Reinhard Laue Universität Bayreuth, Lehrstuhl II für Mathematik, D-95440 Bayreuth, GermanySearch for more papers by this author Reinhard Laue, Reinhard Laue Universität Bayreuth, Lehrstuhl II für Mathematik, D-95440 Bayreuth, GermanySearch for more papers by this author First published: 30 July 1999 https://doi.org/10.1002/(SICI)1520-6610(1999)7:4 3.0.CO;2-ECitations: 6AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinkedInRedditWechat Abstract A halving is a t-design which has the same parameters as its complementary design. Together these two designs form a large set LS[2](t, k, v). There are several recursion theorems for large sets, such that a single new halving results in several new infinite families of halvings. We present new halvings with the parameters 7-(24, 10, 340), 6-(22, 9, 280), 5-(21, 10, 2184), and 5-(21, 9, 910). Recursive constructions by S. Ajoodani-Namini and G. B. Khosrovshahi [Discrete Math 135 (1994), 29–37; J. Combin. Theory A 76 (1996), 139–144] then yield that an LS[2](t, k, v) exists if and only if the parameter set is admissible for t = 6, k = 7, 8, 9, and for t ≤ 5, k ≤ 15. Thus, Hartman's conjecture is true in these cases. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 233–241, 1999 REFERENCES 1 S. Ajoodani-Namini, All block designs with exist, Discrete Math 179 (1998), 27–35. 10.1016/S0012-365X(97)00024-1 Web of Science®Google Scholar 2 S. Ajoodani-Namini, Extending large sets of t-designs, J Combin Theory Ser A 76 (1996), 139–144. 10.1006/jcta.1996.0093 Web of Science®Google Scholar 3 S. Ajoodani-Namini and G. B. Khosrovshashi, More on halving the complete designs, Discrete Math 135 (1994), 29–37. 10.1016/0012-365X(93)E0096-M Web of Science®Google Scholar 4 W. O. Alltop, Extending t-designs, J Combin Theory Ser A 12 (1975), 177–186. 10.1016/0097-3165(75)90006-0 Web of Science®Google Scholar 5 A. Betten, R. Laue, and A. Wassermann, Simple 7-designs with small parameters, to appear in J Combin Designs. Google Scholar 6 A. Betten, A. Kerber, R. Laue, and A. Wassermann, Simple 8-designs with small parameters, to appear in Designs Codes and Cryptography. Google Scholar 7 A. Betten, R. Laue, and A. Wassermann, DISCRETA—-A tool for constructing t-designs, Lehrstuhl II für Mathematik, Universität Bayreuth, Software package and documentation available under httpd://www.mathe2.uni-bayreuth.de/betten/DISCRETA/Index.html Google Scholar 8 A. Betten, R. Laue, and A. Wassermann, New t-designs and large sets of t-designs, to appear in Discrete Math. Google Scholar 9 Y. M. Chee and S. S. Magliveras, A few more large sets of t-designs, preprint. Google Scholar 10 P. B. Gibbons, Computational methods in design theory. The CRC handbook of combinatorial designs, C. J. Colbourn and J. H. Dinitz (Editors), CRC Press, 1996, pp. 718–740. Google Scholar 11 A. Hartman, Halving the complete design, Ann Discrete Math 34 (1987), 207–224. Google Scholar 12 G. B. Khosrovshashi and S. Ajoodani-Namini, Combining t-designs, J Combin Theory A 58 (1991), 26–34. 10.1016/0097-3165(91)90071-N Web of Science®Google Scholar 13 D. L. Kreher and S. P. Radziszowski, The existence of simple 6-(14, 7, 4) designs, J Comb Theory Ser A 43 (1986), 237–243. 10.1016/0097-3165(86)90064-6 Web of Science®Google Scholar 14 D. L. Kreher, An infinite family of (simple) 6-designs, J Combin Des 1 (1993), 277–280. 10.1002/jcd.3180010403 Google Scholar 15 L. Teirlinck, Locally trivial t-designs and t-designs without repeated blocks Discrete Math 77 (1989), 345–356. 10.1016/0012-365X(89)90372-5 Web of Science®Google Scholar 16 Tran van Trung, On the construction of t-designs and the existence of some new infinite families of simple 5-designs, Arch Math 47 (1986), 187–192. 10.1007/BF01193690 Web of Science®Google Scholar 17 A. Wassermann, Finding simple t-designs with enumeration techniques, To appear in J Combin Designs. Google Scholar 18 S. Wolfram, Geometry of binomial coefficients, Amer Math Soc Monthly (November 1984), 566–571. 10.2307/2323743 Web of Science®Google Scholar 19 Qui-Rong Wu, A note on extending t-designs, Australas J Combin 4 (1991), 229–235. Google Scholar Citing Literature Volume7, Issue41999Pages 233-241 ReferencesRelatedInformation