Ternary codes from primitive representations of the group PSL2(9) and a new 2-(15,7,36) design

In this paper, we construct, using computations withMagma, a ternary code C from a primitive permutation representation of degree 15 of the group PSL₂(9) by Key-Moori Method 1. The code C is an optimal code invariant under the group S₆. We consider the action of the automorphism group S₆ on C and its dual. By taking the support of any codeword ? of weight l and orbiting it under S₆, 1-(15, l, k_l) designs are obtained, where k_l = l|?^S₆ |/15. For any codeword, the structure of the stabilizer in S₆ is determined and primitivity of S₆ on each design is examined. It is shown that the complement of one of these designs is actually a new design D with parameters 2-(15, 7, 36). Moreover, Aut(D) ? S₆.