Ternary codes from primitive representations of the group $PSL_2(9)$ and a new 2-(15,7,36) design

Abstract

In this paper, we construct, using computations with Magma, a ternary code $C$ from a primitive permutation representation of degree 15 of the group $PSL_2(9)$ by Key-Moori Method 1. The code $C$ is an optimal code invariant under the group $S_6$. We consider the action of the automorphism group $S_6$ on $C$ and its dual. By taking the support of any codeword $\omega$ of weight $l$ and orbiting it under $S_6$, $1-(15, l, k_l)$ designs are obtained, where $k_l = l|\omega^{S_6}|/15$. For any codeword, the structure of the stabilizer in $S_6$ is determined and primitivity of $S_6$ 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, $Aut(D)\cong S_6$