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On block-transitive designs with affine automorphism group. (English) Zbl 0876.05009

Summary: Detailed necessary and sufficient conditions for a \(k\)-subset of \(\text{AG}(d,3)\) to generate the block set of a block-transitive \(t\)-design with automorphism group \(\text{AGL}(d,3)\) are derived for \(t= 3, 4, 5\). Similar necessary conditions are found for the existence of a block-transitive design with automorphism group \(\text{AGL}(d,p)\) when \(p\) is an arbitrary odd prime. A search was carried out to find feasible parameter sets satisfying the implied divisibility conditions. The only ‘small’ feasible parameter sets found with \(k\) or \(v-k\) not exceeding 1000 were for \(t= 4\) and \((d,p)= (7,3)\), \((8,3)\), and \((3,7)\). Examples of block-transitive 4-designs admitting \(\text{AGL}(7,3)\) are found for each of the values \(k=115, 116, 230, 437\), and \(552\).

MSC:

05B05 Combinatorial aspects of block designs
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
51E05 General block designs in finite geometry

Software:

Maple; Tabu search
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Full Text: DOI

References:

[1] Alltop, W. O., 5-designs in affine spaces, Pacific J. Math., 39, 547-551 (1971) · Zbl 0239.05012
[2] Cameron, P. J., Finite simple groups and finite permutation groups, Bull. London Math. Soc., 13, 1-22 (1981) · Zbl 0463.20003
[3] Cameron, P. J.; Praeger, C. E., Block-transitive \(t\)-designs, I: point-imprimitive designs, Discrete Math., 118, 33-43 (1993) · Zbl 0780.05006
[4] Cameron, P. J.; Praeger, C. E., Block-transitive \(t\)-designs, II: large \(t\), (De Clerck, F.; etal., Finite Geometry and Combinatorics. Finite Geometry and Combinatorics, London Mathematical. Society. Lecture Notes in Mathematics, Vol, 191 (1993), Cambridge University Press: Cambridge University Press Cambridge), 103-119 · Zbl 0792.05017
[5] Char, B. W.; Geddes, K. O.; Gonnet, G. H.; Monagan, M. B.; Watt, S. M., MAPLE reference manual (1990), Waterloo Maple Publications: Waterloo Maple Publications Waterloo · Zbl 0758.68038
[7] Glover, F. E.; Taillard; de Werra, D., A user’s guide to tabu search, Oper. Res., 41, 3-28 (1993) · Zbl 0772.90063
[8] Hughes, D. R.; Piper, F. C., Design Theory (1985), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0561.05009
[9] Praeger, C. E., Block-transitive designs and maximal subgroups, Australas. J. Combin., 1, 193-205 (1990) · Zbl 0776.05012
[10] Ray-Chaudhuri, D. K.; Wilson, R. M., On \(t\)-designs, Osaka J. Math., 12, 737-744 (1975) · Zbl 0342.05018
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