Reza Kahkeshani - Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, Kashan, Iran

In this paper, we use the Key-Moori Method ۱ and construct a quaternary code \mathcal{C}_۸ from a primitive representation of the group PSL_۲(۹) of degree ۱۵. We see that \mathcal{C}_۸ is a self-orthogonal even code with the automorphism group isomorphic to the alternating group A_۸. It is shown that by taking the support of any codeword \omega of weight l in \mathcal{C}_۸ or \mathcal{C}_۸^\bot, and orbiting it under A_۸, a ۲-(۱۵,l,\lambda) design invariant under the group A_۸ is obtained, where \lambda=\binom{l}{۲}|\omega^{A_۸}|/\binom{۱۵}{۲}. A number of these designs have not been known before up to our best knowledge. The structure of the stabilizers (A_۸)_\omega is determined and moreover, primitivity of A_۸ on each design is examined.

Design, Code, Automorphism group, Projective special linear group, Primitive permutation representation