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Self-orthogonal codes from orbit matrices of Seidel and Laplacian matrices of strongly regular graphs
Designs from maximal subgroups and conjugacy classes of Ree groups
1. | School of Mathematical Sciences, North-West University, (Mafikeng) 2754, South Africa |
2. | School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal, Durban 4000, South Africa |
In this paper, using a method of construction of $ 1 $-designs which are not necessarily symmetric, introduced by Key and Moori in [
References:
[1] |
E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics, 103, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9781316529836. |
[2] |
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, Oxford University Press, Eynsham, 1985.
![]() ![]() |
[3] |
I. M. Isaacs, Character Theory of Finite Groups, Dover Publications, Inc., New York, 1994. |
[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] |
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.
|
[6] |
J. D. Key, J. Moori and B. G. Rodrigues,
On some designs and codes from primitive representations of some finite simple groups, Combin. Math. Combin. Comput., 45 (2003), 3-19.
|
[7] |
O. H. King, The subgroup structure of finite classical groups in terms of geometric configurations, in Surveys in Combinatorics, London Math. Soc. Lecture Note Ser., 327, Cambridge
Univ. Press, Cambridge, 2005, 26-56.
doi: 10.1017/CBO9780511734885.003. |
[8] |
V. M. Levchuk and Y. N. Nuzhin,
The structure of Ree groups, Algebra i Logika, 24 (1985), 26-41.
|
[9] |
J. Moori, Finite groups, designs and codes, in Information Security, Coding Theory and Related Combinatorics, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 29, IOS, Amsterdam, 2011,202-230. |
[10] |
J. Moori and B. G. Rodrigues,
A self-orthogonal doubly even code invariant under McL:2, J. Combin. Theory Ser. A, 110 (2005), 53-69.
doi: 10.1016/j.jcta.2004.10.001. |
[11] |
J. Moori and B. G. Rodrigues,
Some designs and codes invariant under the simple group Co2, J. Algebra, 316 (2007), 649-661.
doi: 10.1016/j.jalgebra.2007.02.004. |
[12] |
J. Moori and B. G. Rodrigues,
A self-orthogonal doubly-even code invariant under the McL, Ars Combin., 91 (2009), 321-332.
|
[13] |
J. Moori and B. G. Rodrigues,
On some designs and codes invariant under the Higman-Sims group, Util. Math., 86 (2011), 225-239.
|
[14] |
J. Moori, B. G. Rodrigues, A. Saeidi and S. Zandi,
Some symmetric designs invariant under the small Ree groups, Comm. Algebra, 47 (2019), 2131-2148.
doi: 10.1080/00927872.2018.1530245. |
[15] |
J. Moori and A. Saeidi,
Some designs and codes invariant under the Tits group, Adv. Math. Commun., 11 (2017), 77-82.
doi: 10.3934/amc.2017003. |
[16] |
J. Moori and A. Saeidi,
Some design invariant under the Suzuki groups, Util. Math., 109 (2018), 105-114.
|
[17] |
J. Moori and A. Saeidi,
Constructing some design invariant under the PSL2(q), q even, Comm. Algebra, 46 (2018), 160-166.
doi: 10.1080/00927872.2017.1316854. |
[18] |
R. Ree,
A family of simple groups asssociated with the simple Lie algebra of type (G2), Amer. J. Math, 83 (1961), 432-462.
doi: 10.2307/2372888. |
[19] |
D. O. Revin and E. P. Vdovin,
On the number of classes of conjugate Hall subgroups in finite simple groups, J. Algebra, 324 (2010), 3614-3652.
doi: 10.1016/j.jalgebra.2010.09.014. |
[20] |
H. Ward,
On Ree's series of simple groups, Trans. Amer. Math. Soc., 121 (1966), 62-89.
doi: 10.2307/1994333. |
[21] |
R. A. Wilson, The Finite Simple Groups, Graduate Texts in Mathematics, 251, Springer-Verlag London, Ltd., London, 2009.
doi: 10.1007/978-1-84800-988-2. |
[22] |
R. A. Wilson,
Another new approach to the small Ree groups, Arch. Math. (Basel), 94 (2010), 501-510.
doi: 10.1007/s00013-010-0130-4. |
show all references
References:
[1] |
E. F. Assmus Jr. and J. D. Key, Designs and Their Codes, Cambridge Tracts in Mathematics, 103, Cambridge University Press, Cambridge, 1992.
doi: 10.1017/CBO9781316529836. |
[2] |
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, Oxford University Press, Eynsham, 1985.
![]() ![]() |
[3] |
I. M. Isaacs, Character Theory of Finite Groups, Dover Publications, Inc., New York, 1994. |
[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] |
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.
|
[6] |
J. D. Key, J. Moori and B. G. Rodrigues,
On some designs and codes from primitive representations of some finite simple groups, Combin. Math. Combin. Comput., 45 (2003), 3-19.
|
[7] |
O. H. King, The subgroup structure of finite classical groups in terms of geometric configurations, in Surveys in Combinatorics, London Math. Soc. Lecture Note Ser., 327, Cambridge
Univ. Press, Cambridge, 2005, 26-56.
doi: 10.1017/CBO9780511734885.003. |
[8] |
V. M. Levchuk and Y. N. Nuzhin,
The structure of Ree groups, Algebra i Logika, 24 (1985), 26-41.
|
[9] |
J. Moori, Finite groups, designs and codes, in Information Security, Coding Theory and Related Combinatorics, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., 29, IOS, Amsterdam, 2011,202-230. |
[10] |
J. Moori and B. G. Rodrigues,
A self-orthogonal doubly even code invariant under McL:2, J. Combin. Theory Ser. A, 110 (2005), 53-69.
doi: 10.1016/j.jcta.2004.10.001. |
[11] |
J. Moori and B. G. Rodrigues,
Some designs and codes invariant under the simple group Co2, J. Algebra, 316 (2007), 649-661.
doi: 10.1016/j.jalgebra.2007.02.004. |
[12] |
J. Moori and B. G. Rodrigues,
A self-orthogonal doubly-even code invariant under the McL, Ars Combin., 91 (2009), 321-332.
|
[13] |
J. Moori and B. G. Rodrigues,
On some designs and codes invariant under the Higman-Sims group, Util. Math., 86 (2011), 225-239.
|
[14] |
J. Moori, B. G. Rodrigues, A. Saeidi and S. Zandi,
Some symmetric designs invariant under the small Ree groups, Comm. Algebra, 47 (2019), 2131-2148.
doi: 10.1080/00927872.2018.1530245. |
[15] |
J. Moori and A. Saeidi,
Some designs and codes invariant under the Tits group, Adv. Math. Commun., 11 (2017), 77-82.
doi: 10.3934/amc.2017003. |
[16] |
J. Moori and A. Saeidi,
Some design invariant under the Suzuki groups, Util. Math., 109 (2018), 105-114.
|
[17] |
J. Moori and A. Saeidi,
Constructing some design invariant under the PSL2(q), q even, Comm. Algebra, 46 (2018), 160-166.
doi: 10.1080/00927872.2017.1316854. |
[18] |
R. Ree,
A family of simple groups asssociated with the simple Lie algebra of type (G2), Amer. J. Math, 83 (1961), 432-462.
doi: 10.2307/2372888. |
[19] |
D. O. Revin and E. P. Vdovin,
On the number of classes of conjugate Hall subgroups in finite simple groups, J. Algebra, 324 (2010), 3614-3652.
doi: 10.1016/j.jalgebra.2010.09.014. |
[20] |
H. Ward,
On Ree's series of simple groups, Trans. Amer. Math. Soc., 121 (1966), 62-89.
doi: 10.2307/1994333. |
[21] |
R. A. Wilson, The Finite Simple Groups, Graduate Texts in Mathematics, 251, Springer-Verlag London, Ltd., London, 2009.
doi: 10.1007/978-1-84800-988-2. |
[22] |
R. A. Wilson,
Another new approach to the small Ree groups, Arch. Math. (Basel), 94 (2010), 501-510.
doi: 10.1007/s00013-010-0130-4. |
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