Some Ternary Codes Invariant under the Group $M^{c}L:2$

Abstract

We examine designs ${\mathcal{D}}_i$ and ternary codes $C_i$, where $i\in \{112,113,162,163,274\}$, constructed from a primitive permutation representation of degree 275 of the sporadic simple group $M^{c}L$. We prove that $\dim (C_{113})=22,\dim (C_{162})=21,C_{113}\supset C_{162}$ and $M^{c}L:2$ acts irreducibly on $C_{162}$. Furthermore, we have $C_{112}=C_{163}=C_{274}=V_{{27}_5}(GF(3)),$ $\text{Aut}({\mathcal{D}}_{112})=\text{Aut}({\mathcal{D}}_{163})$ = $\text{Aut}({\mathcal{D}}_{113})=\text{Aut}({\mathcal{D}}_{162})=\text{Aut}(C_{113})=\text{Aut}(C_{162})=M^{c}L:2$ while $Aut({\mathcal{D}}_{274})=Aut(C_{112})=Aut(C_{163})=Aut(C_{274})=S_{275}.$
We also determine the weight distributions of $C_{113}$ and $C_{162}$ and that of their duals.