Quaternary codes and a class of 2-designs invariant under the group $A_8$

In this paper, we use the Key-Moori Method 1 and construct a quaternary code $mathcal{C}_8$ from a primitive representation of the group $PSL_2(9)$ of degree 15. We see that $mathcal{C}_8$ is a self-orthogonal even code with the automorphism group isomorphic to the alternating group $A_8$. It is shown that by taking the support of any codeword $omega$ of weight $l$ in $mathcal{C}_8$ or $mathcal{C}_8^bot$, and orbiting it under $A_8$, a 2-$(15,l,lambda)$ design invariant under the group $A_8$ is obtained, where $lambda=binom{l}{2}|omega^{A_8}|/binom{15}{2}$. A number of these designs have not been known before up to our best knowledge. The structure of the stabilizers $(A_8)_omega$ is determined and moreover, primitivity of $A_8$ on each design is examined.