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Ternary codes from primitive representations of the group PSL2(9) and a new 2-(15,7,36) design
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In this paper, we construct, using computations withMagma, a ternary code C from a primitive permutation representation of degree 15 of the group PSL2(9) by Key-Moori Method 1. The code C is an optimal code invariant under the group S6. We consider the action of the automorphism group S6 on C and its dual. By taking the support of any codeword ? of weight l and orbiting it under S6, 1-(15, l, kl) designs are obtained, where kl = l|?S6 |/15. For any codeword, the structure of the stabilizer in S6 is determined and primitivity of S6 on each design is examined. It is shown that the complement of one of these designs is actually a new design D with parameters 2-(15, 7, 36). Moreover, Aut(D) ? S6.
Keywords
- E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, Cambridge University Press, 1992. (2nd printing with corrections, 1993).
- W. Bosma and J. Cannon, Handbook of Magma Functions, University of Sydney, 1994. Available online at http://www.magma.maths.usyd.edu.au/magma/.
- C. J. Colbourn and J. H. Dinitz, Handbook of Combinatorial Designs, 2nd ed., Chapman & Hall/CRC, 2007.
- J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
- M. R. Darafsheh, Designs from the group PSL2(q), q even, Des. Codes Cryptogr., 39 (2006), 311–316.
- M. R. Darafsheh, A. R. Ashrafi and M. Khademi, Some designs related to group actions, Ars. Combin., 86 (2008), 65–75.
- M. R. Darafsheh, A. R. Ashrafi and M. Khademi, On designs constructed by group actions, J. Combin. Math. Combin. Comput., 70 (2009), 235–245.
- M. R. Darafsheh, A. Iranmanesh and R. Kahkeshani, Some designs invariant under the groups S8 and A9, Des. Codes Cryptogr., 51 (2009), 211–223.
- M. R. Darafsheh, A. Iranmanesh and R. Kahkeshani, Designs from the groups PSL2(q) for certain q, Quaestiones Mathematicae, 32(3) (2009), 297–306.
- R. Kahkeshani, M. R. Darafsheh and A. Iranmanesh, Self-dual designs and their binary codes obtained from the group PSL2(13), Creat. Math. Inform., 18(2) (2009), 172–181.
- M. Grassl, Bounds on the minimum distance of linear codes, Available online at http://www.codetables.de
- J. D. Key and J. Moori, Designs, codes and graphs from the Janko groups J1 and J2, J. Combin. Math. Combin. Comput., 40 (2002), 143–159.
- J. D. Key and J. Moori, Correction to: “Designs, codes and graphs from the Janko groups J1 and J2, J. Combin. Math. Combin. Comput., 40 (2002), 143–159”, J. Combin. Math. Combin. Comput., 64 (2008), 153.
- J. Moori and A. Saeidi, Constructing some designs invariant under PSL2(q), q even, Commun. Alg., 46(1) (2018), 160–166.
- M. Suzuki, Group Theory I, Springer-Verlag, 1982.
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