Some designs and codes from $L_2(q)$

Document Type: Research Paper


1 North-West University (Mafikeng Campus)

2 North-West University


‎For $q \in \{7,8,9,11,13,16\}$‎, ‎we consider the primitive actions of $L_2(q)$ and use Key-Moori Method 1 as described in [Codes‎, ‎designs and graphs from the Janko groups {$J_1$} and‎ ‎{$J_2$}‎, J‎. ‎Combin‎. ‎Math‎. ‎Combin‎. ‎Comput.‎, ‎40 (2002) 143-159.‎, ‎Correction to‎: ‎``Codes‎, ‎designs and graphs from the Janko groups‎ ‎{$J_1$} and {$J_2$}'' [J‎. ‎Combin‎. ‎Math‎. ‎Combin‎. ‎Comput‎., 40 (2002) 143-159]‎, J‎. ‎Combin‎. ‎Math‎. ‎Combin‎. ‎Comput.‎, 64} (2008) 153.] to construct designs from the orbits of the point stabilisers and from any union of these orbits‎. ‎We also use Key-Moori Method 2‎ ‎(see Information security‎, ‎coding theory and related combinatorics‎, ‎NATO Sci‎. ‎Peace Secur‎. ‎Ser‎. ‎D Inf‎. ‎Commun‎. ‎Secur.‎, ‎IOS Amsterdam‎, 29 (2011) 202-230.) to determine the designs from the maximal subgroups and the conjugacy classes of elements of these groups‎. ‎The incidence matrices of these designs are then used to generate associated binary codes‎. ‎The full automorphisms of these designs and codes are also determined‎.


Main Subjects

W. Bosma, J. Cannon and C. Playoust (1997). The Magma algebra system. {I}. {T}he user language. J. Symbolic Comput.. 24 (3-4), 235-265
J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson (1985). Atlas of finite groups. Oxford University Press, Eynsham.
M. R. Darafsheh, A. R. Ashrafi and M. Khademi (2008). Some designs related to group actions. Ars Combin.. 86, 65-75
M. R. Darafsheh, A. Iranmanesh and R. Kahkeshani (2009). Some designs and codes invariant under the groups S_9 and A_8. Des. Codes Cryptogr.. 51 (2), 211-223
J. D. Key and J. Moori (2002). Codes, designs and graphs from the Janko groups J_1 and J_2. J. Combin. Math. Combin. Comput.. 40, 143-159
J. D. Key and J. Moori (2008). Correction to: ``Codes, designs and graphs from the Janko groups J_1 and J_2''. [J. Combin. Math. Combin. Comput. {bf 40} (2002) 143--159], J. Combin. Math. Combin. Comput.. 64
T. Le and J. Moori (2013). On the automorphisms of designs constructed from finite simple groups. Submitted.
J. Moori (2011). Finite groups, designs and codes. Information security, coding theory and related combinatorics, NATO Sci. Peace Secur. Ser. D Inf. Commun. Secur., IOS Amsterdam. 29, 202-230
B. G. Rodrigues (2002). Codes of Designs and Graphs from Finite Simple Groups. PhD thesis, University of Natal.