Open Access2-modular representations of the alternating group A_8 as binary codes
Author(s) -
Lucy Chikamai,
Jamshid Moori,
Bernardo Gabriel Rodrigues
Publication year - 2012
Publication title -
glasnik matematički
Language(s) - English
Resource type - Journals
eISSN - 1846-7989
pISSN - 0017-095X
DOI - 10.3336/gm.47.2.01
Subject(s) - mathematics , modular design , alternating group , binary number , group (periodic table) , modular group , combinatorics , discrete mathematics , arithmetic , symmetric group , computer science , programming language , physics , quantum mechanics
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