Glasnik Matematicki, Vol. 47, No. 2 (2012), 225252.
2MODULAR REPRESENTATIONS OF THE ALTERNATING GROUP A_{8} AS BINARY CODES
L. Chikamai, Jamshid Moori and B. G. Rodrigues
School of Mathematical Sciences , University of KwaZuluNatal , Durban 4041, South Africa
email: chikamail@ukzn.ac.za, luciechikamai@yahoo.com
School of Mathematical Sciences , NorthWest University (Mafikeng) , Mmabatho 2735, South Africa
email: Jamshid.Moori@nwu.ac.za
School of Mathematical Sciences , University of KwaZuluNatal , Durban 4041, South Africa
email: rodrigues@ukzn.ac.za
Abstract. Through a modular representation theoretical approach we enumerate all nontrivial codes from the 2modular representations of A_{8}, using a chain of maximal submodules of a permutation module induced by the action of A_{8} 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 nontrivial codewords in the codes examined constitute single orbits of the automorphism groups that are stabilized by maximal subgroups. Many selforthogonal codes invariant under A_{8} are obtained, and moreover, 22 optimal codes all invariant under A_{8} are constructed. Finally, we establish that there are no selfdual codes of lengths 28 and 56 invariant under A_{8} and S_{8} respectively, and in particular no selfdual doublyeven code of length 56.
2010 Mathematics Subject Classification.
05B05, 20D45, 94B05.
Key words and phrases. Derived, symmetric and quasisymmetric designs, selforthogonal 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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