Kirkman Triple Systems of Orders \(27, 33\), and \(39\)

In the search for doubly resolvable Kirkman triple systems of order \(v\), systems admitting an automorphism of order \((v-3)/3\) fixing three elements, and acting on the remaining elements in three orbits of length \((v-3)/3\), have been of particular interest. We have established by computer that 100 such Kirkman triple systems exist for \(v=21\), 90,598 for \(v=27\), at least 4,494,390 for \(v=33\), and at least 1,626,684 for \(v=39\). This improves substantially on known lower bounds for numbers of Kirkman triple systems. We also establish that the KTS(27)s so produced yield 47 nonisomorphic doubly resolved KTS(27)s admitting the same automorphism.