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, q^{2n+1} + 1) exists.

Suppose q is a prime power and n ≥ 1. Then an S(2, q + 1, q^{2n+2} + q^{2n+1} + 1) exists.

Suppose q is a prime power and n ≥ 1. Then an S(2, q + 1, q^{2n+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

Return to the HCD new results home page.