Truncated Transversal Designs: A New Lower Bound on the Number of Idempotent MOLS of Side n

A truncated transversal design TTD of type gkm1 is a {k, k+1}-GDD of type gkm1 in which each point on the group of size m lies only in blocks of size k+1. Thus a TTD of type gkm1 is equivalent to a transversal design TD (k, g) having m disjoint parallel classes of blocks. We employ a new construction developed by the author (1993, J. Combin. Des.1, 15–26) to show that if g1<g2 and if there exists a TD (k, g1) and a TD (k+1, g2), then there exists a TTD of type (g1g2)km1 for any 0⩽m⩽(g2 div g1) g21. As a corollary, we obtain a new lower bound on the number of mutually orthogonal idempotent latin squares of side g: if g1<g2 and there exist r MOLS of side g1 and r+1 MOLS of side g2 , then N(1 g1g2)⩾r.