Handbook of Combinatorial Designs, Second Edition

New Results in Part II

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 7/5/12

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

Page 39, Design #317. "Nr" is now "≥" 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. 17 (2010), #R139. Marco Buratti (buratti@dmi.unipg.it), June 2012

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, Series A 114 (2007), 235-252. Tony Forbes (tonyforbes@ltkz.demon.co.uk) Jan. 2007.

Page 88-93, Table 4.46. The following list of new 5-designs have been found: 5-(24,9,λ), λ=18, 24, 30 ; 5-(25,9,30); 5-(26,10,126); 5-(32,6,λ), λ=6, 9, 12; 5-(33,7,84); 5-(18,8,λ), λ=12, 18; 5-(19,9,λ), λ=42, 63; 5-(25,10,λ), λ=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. 18, 2010.

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

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. Östergård and Olli Pottonen, Journal of Combinatorial Theory. Series A 115 (2008), 1570-1573, DOI 10.1016/j.jcta.2008.04.005. Olli Pottonen (olli.pottonen@tkk.fi) July 2007.

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. 17 (2010), #R139. Marco Buratti (buratti@dmi.unipg.it), June 2012

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

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