Abstract
We show the existence of simple 8-(31,10,93) and 8-(31,10,100) designs. For each value of λ we show 3 designs in full detail. The designs are constructed with a prescribed group of automorphisms PSL(3,5) using the method of Kramer and Mesner KramerMesner76. They are the first 8-designs with small parameters which are known explicitly. We do not yet know if PSL(3,5) is the full group of automorphisms of the given designs. There are altogether 138 designs with λ = 93 and 1658 designs with λ = 100 and PSL(3,5) as a group of automorphisms. We prove that they are all pairwise non-isomorphic. For this purpose, a brief account on the intersection numbers of these designs is given. The proof is done in two different ways. At first, a quite general group theoretic observation shows that there are no isomorphisms. In a second approach we use the block intersection types as invariants, they classify the designs completely.
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References
Thomas Beth, Dieter Jungnickel, and Hanfried Lenz, Design theory, Cambridge University Press, Cambridge, 1986.
Anton Betten et al., DISCRETA - a tool for constructing t-designs, Lehrstuhl II für Mathematik, Universität Bayreuth, 1997.
Anton Betten, Adalbert Kerber, Axel Kohnert, Reinhard Laue, and Alfred Wassermann, “The discovery of simple 7-designs with automorphism group PΓL(2, 32),” In Applied algebra, algebraic algorithms and error-correcting codes (Paris, 1995), Lecture Notes in Comput. Sci., Vol. 948 (1995) pp. 131–145.
Anton Betten, Reinhard Laue, and Alfred Wassermann, Simple 6 and 7-designs on 19 to 33 points, Congressus Numerantium, Vol. 123 (1997) pp. 149–160.
Anton Betten, Reinhard Laue, and Alfred Wassermann, Some simple 7-designs, In M.J. de Resmini J.W.P. Hirschfeld, S.S. Magliveras, editor, Geometry, Combinatorial Designs and Related Structures, Proceedings of the First Pythagorean Conference, London Mathematical Society Lecture Notes, Cambridge University Press, Cambridge, Vol. 245 (1997) pp. 15–25.
Anton Betten and Alfred Wassermann, {0, 1}-solutions of integer linear equation systems. In Parallel Virtual Machine - EuroPVM'96 (Munich, 1996), Vol. 1156 of Lecture Notes in Comput. Sci., pp. 311–314. Springer, Berlin, (1995).
Egmont Köhler, Allgemeine Schnittzahlen in t-designs, In Proceedings of the Oberwolfach Meeting “Kombinatorik” (1986), Vol. 73, pp. 133–142, (1988/89).
Earl S. Kramer and Dale M. Mesner, t-designs on hypergraphs. Discrete Math., Vol. 15(3) (1976) pp. 263–296.
D. L. Kreher, t-designs, t ≥ 3, In The CRC handbook of combinatorial designs, pp. 47–66. CRC Press, Boca Raton, FL, (1996).
N. S. Mendelsohn, Intersection numbers of t-designs, In Studies in Pure Mathematics (Presented to Richard Rado), pp. 145–150. Academic Press, London, (1971).
Ray-Chaudhuri, Dijen K. and Wilson, Richard M., On t-designs., Osaka J. Math., Vol. 12 (1975) pp. 737–744.
Bernd Schmalz, t-Designs zu vorgegebener Automorphismengruppe, Bayreuth. Math. Schr., Vol. 41 (1992) pp. 1–164. Dissertation, Universität Bayreuth, Bayreuth, 1992.
Bernd Schmalz, The t-designs with prescribed automorphism group, new simple 6-designs, J. Combin. Des., Vol. 1(2) (1993) pp. 125–170.
Stanton, R.G. and Sprott, D.A., Block intersections in balanced incomplete block designs, Can. Math. Bull., Vol. 7 (1964) pp. 539–548.
Vladimir D. Tonchev, Combinatorial configurations: designs, codes, graphs, volume 40 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow, (1988). Translated from the Bulgarian by Robert A. Melter.
Tran van Trung, Qiu-rong Wu, and Dale M. Mesner, High order intersection numbers of t-designs, J. Statist. Plann. Inference, Vol. 56(2) (1996) pp. 257–268. Special issue on orthogonal arrays and affine designs, Part II.
Alfred Wassermann, Finding simple t-designs with enumeration techniques, J. of Combinatorial Designs, Vol. 6(2) (1998) pp. 79–90.
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Betten, A., Kerber, A., Laue, R. et al. Simple 8-Designs with Small Parameters. Designs, Codes and Cryptography 15, 5–27 (1998). https://doi.org/10.1023/A:1008263724078
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DOI: https://doi.org/10.1023/A:1008263724078