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

We use the Key-Moori Method ۱ and examine ۱-designs and codes from the representations of the alternating group A_۷. It is shown that a self-dual symmetric ۲-(۳۵,۱۸,۹) design and an optimal even binary [۲۱,۱۴,۴] LCD code are found such that they are invariant under the full automorphism groups S_۸ and S_۷, respectively. Moreover, designs with parameters ۱-(۲۱,l,k_{۱,l}) and ۱-(۳۵,l,k_{۲,l}) are obtained, where \omega is a codeword, l=wt(\omega), k_{۱,l}=l|\omega^{S_۷}|/۲۱ and k_{۲,l}=l|\omega^{S_۷}|/۳۵. It is seen that there exist a ۲-(۲۱,۵,۱۲) design with the full automorphism group S_۷ among these ۱-designs.

Code, design, automorphism group, Alternating group, Primitive permutation representation