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 \( M^cL \). We prove that \( \dim(C_{113}) = 22, \quad \dim(C_{162}) = 21, \quad C_{113} \supset C_{162}\) and \( M^cL:2 \) 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.