{"id":384,"date":"2021-07-26T13:55:47","date_gmt":"2021-07-26T13:55:47","guid":{"rendered":"https:\/\/site.uvm.edu\/jdinitz\/?page_id=384"},"modified":"2021-07-26T15:55:24","modified_gmt":"2021-07-26T15:55:24","slug":"new-results-in-part-ii","status":"publish","type":"page","link":"https:\/\/site.uvm.edu\/jdinitz\/?page_id=384","title":{"rendered":"New Results in Part II"},"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 II of the Handbook.<\/p>\n\n\n\n

Last edited 9\/17\/2019<\/p>\n\n\n\n

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Page 39, Design #143. Nd = 10,374,196,953. Related to this is the result that there are 355,293,682 3-(32,16,7) designs. These were both found as a consequence of the enumeration of the inequivalent Hadamard matrices of order 32 by Kharaghani and Tayfeh Rezaie (see the New Results section for Part V — page 277). Brendan McKay (bdm@cs.anu.edu.au), Feb. 2012<\/p>\n\n\n\n

Page 39, Design #317. “Nr” is now “\u2265” instead of “?”. In fact a (45,5,2)-RBIBD has been found in the paper M. Buratti, J. Yan and C. Wang, From a 1-rotational RBIBD to a partitioned difference family, Electronic J. Combin<\/em>. 17 (2010), #R139. Marco Buratti (buratti@dmi.unipg.it), June 2012<\/p>\n\n\n\n

Page 69, Remark 2.96. In [826] (Forbes, Grannell & Griggs, ‘On 6-sparse Steiner triple systems’) it is proved that there are infinitely many 6-sparse STSs. Indeed, there is a finite set of 6-sparse STS(v)s with v prime and v == 7 (mod 12), but the paper also describes a product construction which generates infinitely many 6-sparse Steiner triple systems from this set. [826] will appear in J. Combin. Theory,<\/em> Series A 114 (2007), 235-252. Tony Forbes (tonyforbes@ltkz.demon.co.uk) Jan. 2007.<\/p>\n\n\n\n

Page 88-93, Table 4.46. The following list of new 5-designs have been found: 5-(24,9,\u03bb), \u03bb=18, 24, 30 ; 5-(25,9,30); 5-(26,10,126); 5-(32,6,\u03bb), \u03bb=6, 9, 12; 5-(33,7,84); 5-(18,8,\u03bb), \u03bb=12, 18; 5-(19,9,\u03bb), \u03bb=42, 63; 5-(25,10,\u03bb), \u03bb=96,120; 5-(30,12,440); 5-(36,12,45*n), 2<=n<=17. Reference: Mutually disjoint designs and new 5-designs derived from groups and codes, M. Araya, M. Harada, V. Tonchev, A. Wassermann, J. Combin. Des<\/em>. 18<\/strong>, 2010.<\/p>\n\n\n\n

Page 103, Theorem 5.11. Here are some new infinite families:
Suppose q is a prime power and n \u2265 1. Then an S(2, q + 1, q2n+1<\/sup> + 1) exists.
Suppose q is a prime power and n \u2265 1. Then an S(2, q + 1, q2n+2<\/sup> + q2n+1<\/sup> + 1) exists.
Suppose q is a prime power and n \u2265 1. Then an S(2, q + 1, q2n+2<\/sup> + q + 1) exists.
Suppose q is a prime power, q \u2265 3, and n \u2265 1. Then an S(2, q, q 2n+2<\/sup> – q 2n+1<\/sup> + q) exists.
Reference: Recurrence relations for Steiner systems with t = 2, James Nechvatal, Ars<\/em> Combinatoria<\/em> (to appear). James Nechvatal (james.nechvatal@nist.gov) 11\/08<\/p>\n\n\n\n

Page 103, Table 5.17. There is no S(4,5,17). So in the table with k=t+1 and n = 13, “there does not exist” equals 4. Reference: There exists no Steiner system S(4,5,17), Patric R.J. \u00d6sterg\u00e5rd and Olli Pottonen, Journal of Combinatorial Theory<\/em>. Series A 115 (2008), 1570-1573, DOI 10.1016\/j.jcta.2008.04.005. Olli Pottonen (olli.pottonen@tkk.fi) July 2007.<\/p>\n\n\n\n

Page 127, Table 7.38. A (45,5,2)-RBIBD has been found in the paper M. Buratti, J. Yan and C. Wang, From a 1-rotational RBIBD to a partitioned difference family, Electronic J. Combin<\/em>. 17 (2010), #R139. Marco Buratti (buratti@dmi.unipg.it), June 2012<\/p>\n\n\n\n

Page 127, Table 7.40. Tripling the (45,5,2)-RBIBD noted above, one obtains a (175,7,6)-RBIBD. Marco Buratti (buratti@dmi.unipg.it), June 2012<\/p>\n\n\n\n

Page 127, Table 7.41.  An  RBIBD(v,8,1) exists for v \\in   {624, 1576, 2976, 5720, 5776, 10200, 14176, 24480}.    S. Costa, T. Feng and  X. Wang,   Des. Codes. Cryptogr<\/em>. 86,  2725-2745 (2018).<\/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 II of the Handbook. Last edited 9\/17\/2019 Page 39, Design #143. Nd = 10,374,196,953. Related to this is the … <\/p>\n

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