Some designs and codes invariant under the groups S9 and A8

We examine some designs and binary codes constructed from the primitive permutation representations of the groups PSL ₂(8) and PSL ₂(9). For PSL ₂(8) of degree 36, we construct a design and its code with the automorphism groups PSL ₂(8) and S ₉, respectively. For PSL ₂(8) of degree 36 and PSL ₂(9) of degree 15, we construct some designs and its codes invariant under the groups S ₉ and A ₈, respectively. The weight distribution and the dual of these codes are determined. By considering the action of automorphism groups on some of these codes, we obtain the structure of the stabilizer for every codeword and construct some designs such that S ₉ or A ₈ act primitively on them.