There are \(270, 474, 142\) nonisomorphic \(2\)-\((9, 4, 6)\) designs (Q2744386)

The author uses an orderly algorithm to determine that there are precisely \(270,474,142\) nonisomorphic 2-\((9,4,6)\) designs. The orderly algorithm constructs the incidence matrices of the designs row-by-row using an algorithm of \textit{P. C. Denny} and \textit{R. Mathon} [A census of \(t\)-\((t+8,t+2,4)\) designs, \(2 \leq t \leq 4\), J. Stat. Plann. Inference, to appear] to check for canonicity of each partial incidence matrix. The enumeration took about three months of total CPU time on 233-500 MHz PC computers, but an elapsed time of only one week by distributing the computation over a network of computers using \textit{B. D. McKay}'s program \textit{autoson} in [``autoson -- a distributed batch system for UNIX workstation networks (version 1.3)'', Tech. Rep. TR-CS-96-03, Computer Science Department, Australian National University, 1996]. Properties of the resulting designs are tabulated according to the size of their automorphism groups. For each group size the numbers of simple and decomposable designs are listed. Finally the canonical incidence matrix of the unique 2-\((9,4,6)\) design with the largest automorphism group (of order \(360\)) is presented. This is another impressive result from an author who has recently cracked the major problem of counting all 11 billion odd \(S(2,3,19)\) designs.