Truncated Transversal Designs: A New Lower Bound on the Number of Idempotent MOLS of Side n
✍ Scribed by Rolf S. Rees
- Publisher
- Elsevier Science
- Year
- 2000
- Tongue
- English
- Weight
- 115 KB
- Volume
- 90
- Category
- Article
- ISSN
- 0097-3165
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✦ Synopsis
A truncated transversal design TTD of type g k m 1 is a [k, k+1]-GDD of type g k m 1 in which each point on the group of size m lies only in blocks of size k+1. Thus a TTD of type g k m 1 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 g 1 <g 2 and if there exists a TD (k, g 1 ) and a TD (k+1, g 2 ), then there exists a TTD of type ( g 1 g 2 ) k m 1 for any 0 m (g 2 div g 1 ) g 2 1 . As a corollary, we obtain a new lower bound on the number of mutually orthogonal idempotent latin squares of side g: if g 1 <g 2 and there exist r MOLS of side g 1 and r+1 MOLS of side g 2 , then N(1 g 1 g 2 ) r.