Abstract
Quasi-symmetric designs are block designs with two block intersection numbersx andy It is shown that with the exception of (x, y)=(0, 1), for a fixed value of the block sizek, there are finitely many such designs. Some finiteness results on block graphs are derived. For a quasi-symmetric 3-design with positivex andy, the intersection numbers are shown to be roots of a quadratic whose coefficients are polynomial functions ofv, k and λ. Using this quadratic, various characterizations of the Witt—Lüneburg design on 23 points are obtained. It is shown that ifx=1, then a fixed value of λ determines at most finitely many such designs.
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References
A. Baartmans andM. S. Shrikhande, Designs with no three mutually disjoint blocks,Discrete Math. 40 (1982), 129–139.
P. J. Cameron, Extending Symmetric designs,J. Combin. Theory (A),14 (1973), 215–220.
P. J. Cameron, Near regularity conditions for designs,Geometriae Dedicata 2 (1973), 213–223.
P. J. Cameron andJ. H. Van Lint,Graphs, Codes and Designs, London Math. Soc. Lecture Note Series,43 (1980).
P. Delsarte, An algebraic approach to the association schemes of coding theory,Philips Res. Repts, Supp. 10 (1973).
A. Hedayat andS. Kageyama, The family oft-designs — Part I,J. Statist. Plann. and Inference 4 (1980), 173–212.
R. Holliday, Quasi-symmetric designs withy=λ,Congressus Numerantum 48 (1985), 195–201.
N. Ito, On tight 4-designs,Osaka J. Math. 12 (1975), 493–522.
S. Kageeyama andA. Hedayat, The family oft-designs — Part II,J. Statist. Plann. and Inference 7 (1983), 257–287.
C. W. H. Lam, L. Thiel, S. Swiercz andJ. McKay, The non-existence of ovals in the projective plane of order 10,Discrete Math. 45 (1983), 319–321.
N. B. Limaye, S. S. Sane andM. S. Shrikhande, The structure of triangle-free quasi-symmetric designs,Discrete Math. 64 (1987), 199–207.
D. K. Ray-Chaudhuri andR. M. Wilson, Ont-designs,Osaka J. of Math. 12 (1975), 737–744.
S. S. Sane andM. S. Shrikhande, Finiteness questions in quasi-symmetric designs,J. Combin. Theory (A),42 (1986), 252–258.
M. S. Shrikhande, Designs with triangle-free graph,in: Proceedings of the seminar on combinatorics and Graph Theory in honor of S. S. Shrikhande at the Indian Statistical Institute, Calcutta, (1982), 334–339.
M. S. Shrikhande, A survey of some problems in combinatorial designs — a matrix approach,Linear Algebra and Appl.,79 (1986), 215–247.
R. M. Wilson, On the theory oft-designs,in: Enumeration and Designs, Academic Press (1984) 19–49.