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 9/02/09
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 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.
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