New results on large sets of Kirkman triple systems

The existence problem on the large sets of Kirkman triple systems (LKTS) was posed by Sylvester in 1850’s as an extension of Kirkman’s 15 schoolgirls problem. An LKTS(15) was constructed by Denniston in 1974. However, up to now the smallest unknown order for the existence of LKTS is still 21. In this paper we construct the two smallest unknown LKTS(v)s with v = 21 and v = 39 by using multiplier automorphism groups. Applying known recursive constructions, we show the existence of more infinite classes of large sets of Kirkman triple systems.