On some designs constructed from the groups $PSL_{2}(q)$, $q=53,61,64$

Document Type : Research Paper

Author

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran

10.29252/as.2020.1718

Abstract

In this paper, we use the primitive permutation representations of the simple groups $PSL_2(53)$, $PSL_2(61)$ and $PSL_2(64)$ and construct 1-designs by the Key-Moori Method 1.
It is shown that the groups $PSL_2(53)$, $PSL_2(53)\text{:}2$, $PSL_2(61)$, $PSL_2(61)\text{:}2$, $PSL_2(64)$, $PSL_2(64)\text{:}2$, $PSL_2(64)\text{:}3$ and $PSL_2(64)\text{:}6$ appear as the full automorphism groups of these obtained designs.

Keywords


[1] E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, Cambridge, (1992).
[2] W. Bosma and J. Cannon, Handbook of Magma Functions, Department of Mathematics, University of
Sydney, (1994), http://www.magma.maths.usyd.edu.au/magma/.
[3] T. Beth, D. Jungnickel and H. Lenz, Design Theory Vol. 1, (second edition), Cambridge University Press, Cambridge, (1999).
[4] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford University Press, Oxford, (1985).
[5] M. R. Darafsheh, Designs from the Group PSL2(q), q Even, Des. Codes Cryptogr. 39 (2006), pp. 311-316.
[6] M. R. Darafsheh, A.R. Ashra and M. Khademi, Some Designs Related to Group Actions, Ars. Combin. 86 (2008), pp. 65-75.
[7] M. R. Darafsheh, A.R. Ashra and M. Khademi, On Designs Constructed by Group Actions, J. Combin. Math. Combin. Comput. 70 (2009), pp. 235-245.
[8] M. R. Darafsheh, A. Iranmanesh and R. Kahkeshani, Designs from the Groups PSL2(q) for Certain q, Quaest. Math. 32 (2009), pp. 1-10.
[9] R. Kahkeshani, 1-Designs from the Group PSL2(59) and Their Automorphism Groups, Math. Interdisc. Res. 3 (2018), pp. 147-158.
[10] J. D. Key and J. Moori, Designs, Dodes and Graphs from the Janko Groups J1 and J2, J. Combin. Math. Combin. Comput. 40 (2002), pp. 143-159.
[11] J. D. Key and J. Moori, Correction to: Codes, Designs and Graphs from the Janko Groups J1 and J2, J. D. Key and J. Moori, JCMCC 40 (2002), pp. 143-159, J. Combin. Math. Combin. Comput. 64 (2008),
pp. 153.
[12] J. D. Key, J. Moori and B.G. Rodrigues, On some Designs and Codes from Primitive Representations of some Finite Simple Groups, J. Combin. Math. Combin. Comput. 45 (2002), pp. 3-19.
[13] J. Moori, Finite Groups, Designs and Codes, Information Security, Coding Theory and Related Combinatorics, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur. Vol. 29 (2011), pp. 202-230.
[14] B. Rodrigues, Codes of Designs and Graphs from Finite Simple Groups, Ph.D. Thesis, University of Natal, South Africa, (2003).
[15] J. J. Rotman, An Introduction to the Theory of Groups, (fourth edition), Springer-Verlag, Berlin, (1995).
[16] M. Suzuki, Group Theory I, Springer-Verlag, Berlin, (1982).