Some designs from the fixed points of alternating groups

Document Type : Research Paper

Authors

1 Department of Mathematics and Applied Mathematics, Faculty of Science and Agriculture, University of Limpopo, South Africa.

2 Department of mathematics and Applied Mathematics, University of Limpopo, South Africa.

3 Department of Pure and Applied Analytics, North West University, mafikeng Campus, Mmabatho, South Africa.

Abstract

In this paper, we construct some $1-(v,k,\lambda)$ designs from the alternating group $G=A_{n}$ with the maximal subgroup isomorphic to $M=A_{n-1}$. The method we use is called Key-Moori Method $2$. Furthermore, from the set $I_x$ which is the intersection of all blocks containing the point $x\in G$, we construct corresponding reduced designs. Our aim is to give explicit formulae to compute the parameters of the designs based on the cyclic structures of the permutations in $G$.

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References

[1] J. D. Key and J. Moori, Designs from maximal subgroups and conjugacy classes of finite simple groups, J. Combin. Math. Combin. Comput., 99 (2016) 41–60.
[2] J. D. Key and J. Moori, Correction to: “Codes, designs and graphs from the Janko groups J1 and J2”, J. Combin. Math. Combin. Comput., 64 (2008) 153.
[3] E. F. Assmus and J. D. Key, Designs and their codes, Cambridge Tracts in Mathematics, 103, Cambridge University Press, Cambridge, 1992.
[4] J. D. Key and J. Moori, Codes, designs and graphs from the Janko groups J1 and J2, J. Combin. Math. Combin. Comput., 40 (2002) 143–159.
[5] T. Le and J. Moori, On the automorphisms of designs constructed from finite simple groups, Des. Codes Cryptogr., 76 no. 3 (2015) 505–517.
[6] J. Moori, Designs and codes from PSL2(q), Group theory, combinatorics, and computing, Contemp. Math., Amer. Math. Soc., Providence, RI, 611 (2014) 137–149.
[7] Moori, J. (2011). Finite groups, designs and codes. In Information Security, Coding Theory and Related Combinatorics (pp. 202-230). IOS Press.
[8] J. Moori and B. G. Rodrigues, On some designs and codes invariant under the Higman-Sims group, Util. Math., 86 (2011) 225–239.
[9] J. Moori and B. G. Rodrigues, Some designs and codes invariant under the simple group Co2, J. Algebra, 316 no. 2 (2007) 649–661.
[10] J. Moori and A. Saeidi, Constructing some designs invariant under PSL2(q), q even, Comm. Algebra, 46 no. 1 (2018) 160–166.
[11] J. Moori and A. Saeidi, Some designs invariant under the Suzuki groups, Util. Math., 109 (2018) 105–114.
[12] A. Saeidi, Reduced designs constructed by Key-Moori Method 2 and their connection with Method 3, AUT J. Math. Comput., 4 no. 1 (2023) 39–46.
[13] A. Saeidi, Designs and codes from fixed points of alternating groups, Comm. Algebra, 50 no. 5 (2022) 2215–2222.
[14] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of finite groups, Maximal subgroups and ordinary characters for simple groups, With computational assistance from J. G. Thackray, Oxford University Press, Eynsham, 1985.

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History

  • Receive Date: 27 May 2024
  • Revise Date: 13 February 2025
  • Accept Date: 15 February 2025
  • Published Online: 18 February 2025

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