An improved product construction for large sets of Kirkman triple systems

It has been shown by Lei, in his recent paper, that there exists a large set of Kirkman triple systems of order uv (LKTS(uv)) if there exist an LKTS(v), a TKTS(v) and an LR(u), where a TKTS(v) is a transitive Kirkman triple system of order v, and an LR(u) is a new kind of design introduced by Lei. In this paper, we improve this product construction by removing the condition “there exists a TKTS(v)”. Our main idea is to use transitive resolvable idempotent symmetric quasigroups instead of TKTS. As an application, we can combine the known results on LKTS and LR-designs to obtain the existence of an LKTS(3nm(2·13n1+1)⋯(2·13nt+1)) for n⩾1, m∈{1,5,11,17,25,35,43,67,91,123}∪{22r+125s+1 : r⩾0,s⩾0}, t⩾0 and ni⩾1 (i=1,…,t).