{"id":395,"date":"2021-07-26T13:56:39","date_gmt":"2021-07-26T13:56:39","guid":{"rendered":"https:\/\/site.uvm.edu\/jdinitz\/?page_id=395"},"modified":"2022-04-13T14:19:15","modified_gmt":"2022-04-13T14:19:15","slug":"new-results-in-part-iv","status":"publish","type":"page","link":"https:\/\/site.uvm.edu\/jdinitz\/?page_id=395","title":{"rendered":"New Results in Part IV"},"content":{"rendered":"\n
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This page contains new results in the area of combinatorial designs that have occurred since the publication of the Handbook of Combinatorial Designs, Second Edition <\/em>in November 2006. The results here would be contained in Part IV of the Handbook.<\/p>\n\n\n\n

Last edited 4\/13\/2022<\/p>\n\n\n\n

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Page 250, Table 3.23. {114, 150, 159, 183, 195, 210} \\subset B({4,7,9}).   H. Wei and G. Ge. Discr. Math<\/em>. 313<\/strong>, 2065-2083 (2013).<\/p>\n\n\n\n

Page 251, Table 3.23.  Since 150 \\in B({4,7,9}), it follows that 150 \\in B({4,7,8,9}).<\/p>\n\n\n\n

Page 251, Table 3.23.  {247, 267} \\subset B({5,6,8}).  H. Yu,   X. Sun,  D. Wu and R.J.R. Abel,   J. Combin. Des<\/em>. 27<\/strong> (2019),   27-41.<\/p>\n\n\n\n

Page 251, Table 3.23.  263 \\in B({5,7}),  H. Yu,   X. Sun,  D. Wu and R.J.R. Abel,   J. Combin. Des<\/em>. 27<\/strong> (2019),   27-41.<\/p>\n\n\n\n

Page 252, Table 3.23. 198 \\in B({5,6,9}),   R.J.R. Abel and    F.E. Bennett,   J. Combin. Des. 13<\/strong> (2005), 239-267.    Also {190, 211} \\subset  B({5,7,8})  H. Yu,   X. Sun,  D. Wu and R.J.R. Abel,   J. Combin. Des.<\/em> 27<\/strong> (2019),   27-41. <\/p>\n\n\n\n

Page 256, Theorem 4.5. This theorem is now a special case of the following theorem: Let u<\/em>, v<\/em> and x<\/em> be positive integers with v<\/em> \u2264 u<\/em>. Then a 3-GDD of type u1<\/sup> v1<\/sup> 1x<\/sup><\/em> exists if and only if u<\/em>, v<\/em> and x<\/em> are odd, uv+ux+vx+\\binom{x}{2} \u2261 0 (mod 3), and x<\/em> \u2265 u<\/em>. This result appears in Darryn Bryant and Daniel Horsley, Steiner triple systems with two disjoint subsystems, J. Combin. Des.<\/em>, 14<\/strong>, 14-24 (2006). Darryn Bryant (db@maths.uq.edu.au) Sept. 2007.<\/p>\n\n\n\n

Page 257, Table 4.10. The three unsolved orders in this table have all been found. An example of a 4-GDD of type 23<\/sup>54<\/sup> and of type 35<\/sup> 62<\/sup> is given in the following reference: R. Julian R. Abel, Yudhistira A. Bunjamin and Diana Combe,\u00a0Some new group divisible designs with block size 4 and two or three group sizes, J. Combin. Des.<\/em>\u00a028<\/strong>\u00a0(2020), 614\u2013628\u00a010.1002\/jcd.21719<\/a>. An example of a 4-GDD of type 22<\/sup> 55<\/sup> is given in the following reference: R. Julian R. Abel, Thomas Britz, Yudhistira A. Bunjamin and Diana Combe,\u00a0Group divisible designs with block size 4 and group sizes 2 and 5, J. Combin. Des.<\/em>\u00a030<\/strong>\u00a0(2022), 367\u2013383.\u00a010.1002\/jcd.21830<\/a>. Yudhi Bunjamin (Yudhi@unsw.edu.au<\/a>) April 2022.<\/p>\n\n\n\n

Page 260, Theorem 4.32. The four possible exceptions with m = 3 and \u03bb = 1, namely (n,t) \\in {(6,14), (6,15), (6,18), (6,23)}, were constructed directly by Cao, Wang and Wei. The result appears in H. Cao, L. Wang, R. Wei, The existence of HGDDs with block size four and its application to double frames, Discrete Mathematics<\/em>, Volume 309, Issue 4 (2009), Pages 945-949. Fei Gao (feigao.chn@gmail.com), March 2010.<\/p>\n\n\n\n

Page 265, Theorem 5.44. The 4-RGDD of type 1227<\/sup> was found in Theorem 2.1 of X. Sun and G. Ge, “Resolvable group divisible designs with block size four and general index,” Discrete Math<\/em>. 309,<\/strong> pp. 2982-2989, 2009. http:\/\/dx.doi.org\/10.1016\/j.disc.2008.07.029. Fei Gao (feigao.chn@gmail.com), August 2010.<\/p>\n\n\n\n

Page 265, Theorem 5.44. 4-RGDD of types 2142<\/sup> , 2346<\/sup>, 654<\/sup> and 2423<\/sup> were found in E. Schuster and G. Ge, “On uniformly resolvable designs with block sizes 3 and 4”, Des. Codes Cryptogr<\/em>., 57<\/strong>, pp 45 — 69, 2010. http:\/\/dx.doi.org\/10.1007\/s10623-009-9348-1. Fei Gao (feigao.chn@gmail.com), August 2010.<\/p>\n\n\n\n

Page 265, Theorem 5.45. The {4,3}-RGDD of type 254<\/sup> was found in Theorem 2.2 of X. Sun and G. Ge, “Resolvable group divisible designs with block size four and general index,” Discrete Math.<\/em> 309<\/strong>, pp. 2982-2989, 2009. http:\/\/dx.doi.org\/10.1016\/j.disc.2008.07.029. Fei Gao (feigao.chn@gmail.com), August 2010.<\/p>\n\n\n\n

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Return to the HCD new results home page.<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

This page contains new results in the area of combinatorial designs that have occurred since the publication of the Handbook of Combinatorial Designs, Second Edition in November 2006. The results here would be contained in Part IV of the Handbook. Last edited 4\/13\/2022 Page 250, Table 3.23. {114, 150, 159, 183, 195, 210} \\subset B({4,7,9}).   … <\/p>\n

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