An improved product construction for large sets of Kirkman triple systems

Abstract

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(3 n m(2·13 n 1 +1)⋯(2·13 n t +1)) for n⩾1, m∈{1,5,11,17,25,35,43,67,91,123}∪{2 2r+125 s+1 : r⩾0,s⩾0} , t⩾0 and n i⩾1 (i=1,…,t) .