Annals of Combinatorics 1 (1997) 107-118
Holey Self-orthogonal Latin Squares with Symmetric Orthogonal Mates
F.E. Bennett, H. Zhang, and L. Zhu
Department of Math., Mount Saint Vincent University, Halifax,
Nova Scotia B3M 2J6, Canada
Department of Computer Science, University of Iowa, Iowa
City, IA 52240, USA
Department of Math., Suzhou University, Suzhou 215006,
Received July 30, 1996
AMS Subject Classification: 05B15
Abstract. We improve the existence results for holey self-orthogonal Latin squares with symme- tric orthogonal mates (HSOLSSOMs) and show that the necessary conditions for the existence of a HSOLSSOM of type hn are also sufficient with at most 28 pairs (h, n) of possible exceptions.
Keywords: Latin square, orthogonal mate, holes, self-orthogonal Latin square
1. R.J.R. Abel, A.E. Brouwer, C.J. Colbourn, and J.H. Dinitz, Mutually orthogonal Latin squares (MOLS), In: CRC Handbook of Combinatorial Designs, C.J. Colbourn and J.H. Dinitz, Eds., CRC Press, Inc., 1996, pp. 111–142.
2. F.E. Bennett, C.J. Colbourn, and R.C. Mullin, Quintessential pairwise balanced designs, J. Statist. Plan. Infer., to appear.
3. F.E. Bennett and L. Zhu, Further results on the existence of HSOLSSOM (hn), Australasian J. Combin. 14 (1996) 207–220.
4. F.E. Bennett and L. Zhu, The spectrum of HSOLSSOM (hn) where h is even, Discrete Math. 158 (1996) 11–25.
5. Th. Beth, D. Jungnickel, and H. Lenz, Design Theory, Bibliographisches Institut, Zurich, 1985.
6. C.C. Lindner, R.C. Mullin, and D.R. Stinson, On the spectrum of resolvable orthogonal arrays invariant under the Klein group K4, Aequationes Math. 26 (1983) 176–183.
7. C.C. Lindner and D.R. Stinson, Steiner pentagon systems, Discrete Math. 52 (1984) 67–74.
8. R.C. Mullin and D.R. Stinson, Holey SOLSSOMs, Utilitas Math. 25 (1984) 159–169.
9. R.C. Mullin and L. Zhu, The spectrum of HSOLSSOM(hn) where h is odd, Utilitas Math. 27 (1985) 157–168.
10. D.R. Stinson and L. Zhu, On the existence of certain SOLS with holes, JCMCC 15 (1994) 33–45.
11. R.M. Wilson, Constructions and uses of pairwise balanced designs, Math. Centre Tracts 55 (1974) 18–41.
12. J. Yin, A.C.H. Ling, C.J. Colbourn, and R.J.R. Abel, The existence of uniform 5-GDDs, J. Combin. Designs, to appear.
13. L. Zhu, Existence for holey SOLSSOM of type 2n, Congressus Numerantium 45 (1984) 295–304.
14. L. Zhu, Existence of three-fold BIBDs with block-size seven, Applied Mathematics – A Journal of Chinese Universities 7 (1992) 321–326.