Glasnik Matematicki, Vol. 47, No. 2 (2012), 225-252.

2-MODULAR REPRESENTATIONS OF THE ALTERNATING GROUP A8 AS BINARY CODES

L. Chikamai, Jamshid Moori and B. G. Rodrigues

School of Mathematical Sciences , University of KwaZulu-Natal , Durban 4041, South Africa
e-mail: chikamail@ukzn.ac.za, luciechikamai@yahoo.com

School of Mathematical Sciences , North-West University (Mafikeng) , Mmabatho 2735, South Africa
e-mail: Jamshid.Moori@nwu.ac.za

School of Mathematical Sciences , University of KwaZulu-Natal , Durban 4041, South Africa
e-mail: rodrigues@ukzn.ac.za

Abstract.   Through a modular representation theoretical approach we enumerate all non-trivial codes from the 2-modular representations of A8, using a chain of maximal submodules of a permutation module induced by the action of A8 on objects such as points, Steiner S(3,4,8) systems, duads, bisections and triads. Using the geometry of these objects we attempt to gain some insight into the nature of possible codewords, particularly those of minimum weight. Several sets of non-trivial codewords in the codes examined constitute single orbits of the automorphism groups that are stabilized by maximal subgroups. Many self-orthogonal codes invariant under A8 are obtained, and moreover, 22 optimal codes all invariant under A8 are constructed. Finally, we establish that there are no self-dual codes of lengths 28 and 56 invariant under A8 and S8 respectively, and in particular no self-dual doubly-even code of length 56.

2010 Mathematics Subject Classification.   05B05, 20D45, 94B05.

Key words and phrases.   Derived, symmetric and quasi-symmetric designs, self-orthogonal designs, codes, optimal linear code, automorphism group, modular representation, alternating group.

Full text (PDF) (free access)

DOI: 10.3336/gm.47.2.01

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