Enumeration of 2-(9, 3, \({\lambda}\)) designs and their resolutions (Q697635)

A design is said to be resolvable if the blocks can be partitioned into parallel classes, each of which partitions the point set. A partition of the blocks into parallel classes is a resolution of the design, and the design is said to be resolved. 2-\((9,3,\lambda)\) designs are known to exist for all \(\lambda \geq 1\). The authors enumerate such designs for \(\lambda = 5\) and their resolutions for \(3 \leq \lambda \leq 5\), the smallest open cases. Two main approaches can be used in such an enumeration. Either the resolutions can be constructed from scratch, or the complete set of 2-\((9,3,\lambda)\) designs can be generated and then checked for resolvability. The authors in fact use both approaches, the first based on an orderly algorithm used by them in checking for the existence of a resolvable \((15,5,4)\) BIBD in [J. Comb. Des. 9, 357-362 (2001; Zbl 1020.05011)], and the second involving a fast maximum clique-finding algorithm developed by the first author in [Discrete Appl. Math. 120, 197-207 (2002; Zbl 1019.05054)]. Summarising the authors' results, the \(395\) resolvable 2-\((9,3,3)\) designs (enumerated by \textit{R. Mathon} and \textit{D. Lomas} in [Australas. J. Comb. 5, 145-158 (1992; Zbl 0770.05009)]) have 426 nonisomorphic resolutions. The authors have confirmed the result of E. Spence (reported in \textit{R. Mathon} and \textit{A. Rosa} [The CRC handbook of combinatorial designs, CRC Press, Boca Raton, 3-41 (1996; Zbl 0845.05008)]) that there are \(16,585,031\) nonisomorphic 2-\((9,3,4)\) designs, and found that there are \(119,985\) nonisomorphic resolvable 2-\((9,3,4)\) designs with \(149,041\) nonisomorphic resolutions. Finally, the number of 2-\((9,3,5)\) nonisomorphic designs is \(5,862,121,434\), and these have \(203,047,732\) nonisomorphic resolutions. As the number of nonisomorphic resolutions was found by constructing resolvable designs from scratch, the authors were not able by this approach to determine the number of 2-\((9,3,5)\) designs that are resolvable. The authors present tables giving the automorphism group sizes for the 2-\((9,3,5)\) designs and for the resolutions of 2-\((9,3,\lambda)\) designs with \(3 \leq \lambda \leq 5\).