A family of 4-designs on 26 points

Acketa, Dragan M. ; Mudrinski, Vojislav

Commentationes Mathematicae Universitatis Carolinae, Tome 37 (1996), p. 843-860 / Harvested from Czech Digital Mathematics Library

Résumé

Using the Kramer-Mesner method, $4$-$(26,6,\lambda)$ designs with $PSL(2,25)$ as a group of automorphisms and with $\lambda$ in the set $\{30,51,60,81,90,111\}$ are constructed. The search uses specific partitioning of columns of the orbit incidence matrix, related to so-called ``quasi-designs''. Actions of groups $PSL(2,25)$, $PGL(2,25)$ and twisted $PGL(2,25)$ are being compared. It is shown that there exist $4$-$(26,6,\lambda)$ designs with $PGL(2,25)$, respectively twisted $PGL(2,25)$ as a group of automorphisms and with $\lambda$ in the set $\{51,60,81,90,111\}$. With $\lambda$ in the set $\{60,81\}$, there exist designs which possess all three considered groups as groups of automorphisms. An overview of $t$-$(q+1,k,\lambda)$ designs with $PSL(2,q)$ as group of automorphisms and with $(t,k) \in \{(4,5), (4,6), (5,6)\}$ is included.

Détail

Publié le : 1996-01-01

MR 1440715 | Zbl 0886.05038

Classification:  05B05,  05B30

Bibliographie

Acketa D.M.; Mudrinski V. A $4$-design on $38$ points, submitted.

Acketa D.M.; Mudrinski V.; Paunić Dj. A search for $4$-designs arising by action of $PGL(2,q)$, Publ. Elektrotehn. Fak., Univ. Beograd, Ser. Mat. 5 (1994), 13-18. (1994) | MR 1322263 | Zbl 0816.05017

Acketa D.M.; Mudrinski V. Two $5$-designs on $32$ points, accepted for Discrete Mathematics. | Zbl 0873.05010

Alltop W.O. An infinite class of $4$-designs, J. Comb. Th. 6 (1969), 320-322. (1969) | MR 0241316 | Zbl 0169.01903

Beth T.; Jungnickel D.; Lenz B. Design Theory, Bibliographisches Institut Mannheim- Wien-Zürich, 1985. | MR 0779284 | Zbl 0945.05005

Chee Y.M.; Colbourn C.J.; Kreher D.L. Simple $t$-designs with $t \leq 30$, Ars Combinatoria 29 (1990), 193-258. (1990) | MR 1046108

Dautović S.; Acketa D.M.; Mudrinski V. A graph approach to isomorphism testing of $4$-$(48,5,\lambda)$ designs arising from $PSL(2,47)$, submitted.

Denniston R.H.F. Some new $5$-designs, Bull. London Math. Soc. 8 (1976), 263-267. (1976) | MR 0480077 | Zbl 0339.05019

Driessen L.M.H.E. $t$-designs, $t \geq 3$, Tech. Report (1978), Department of Mathematics, Eindhover University of Technology, Holland.

Gorenstein D. Finite Simple Groups, An Introduction to Their Classification, Plenum Press, New York, London, 1982. | MR 0698782 | Zbl 0672.20010

Huppert B. Endliche Gruppen, I. Die Grundlehren der matematischen Wissenschaften, Band 134 (1967), Springer-Verlag, Berlin, Heidelberg, New York, xii + 793 pp. | MR 0224703

Huppert B.; Blackburn N. Finite Groups, III., Die Grundlehren der matematischen Wissenschaften, Band 243 (1982), Springer-Verlag, Berlin, Heidelberg, New York, p.454. | MR 0662826 | Zbl 0514.20002

Janko Z.; Tonchev V. Private communication, .

Kramer E.S.; Leavitt D.W.; Magliveras S.S. Construction procedures for $t$-designs and the existence of new simple $6$-designs, Ann. Discrete Math. 26 (1985), 247-274. (1985) | MR 0833794 | Zbl 0585.05002

Kramer E.S.; Mesner D.M. $t$-designs on hypergraphs, Discrete Math. 15 (1976), 263-296. (1976) | MR 0460143 | Zbl 0362.05049

Kreher D.L.; Radziszowski S.P. The existence of simple $6$-$(14,7,4)$ designs, Jour. of Combinatorial Theory, Ser. A 43 (1986), 237-243. (1986) | MR 0867649 | Zbl 0647.05013

Kreher D.L.; Radziszowski S.P. Simple $5$-$(28,6,\lambda)$ designs from $PSL_2(27)$, Ann. Discrete Math. 34 (1987), 315-318. (1987) | MR 0920656