Modules of G-dimension zero over local rings with the cube of maximal ideal being zero

Let $(R,\text{m})$ be a commutative Noetherian local ring with ${\text{m}}^3=(0)$. We give a condition for $R$ to have a non-free module of G-dimension zero. We shall also construct a family of non-isomorphic indecomposable modules of G-dimension zero with parameters in an open subset of projective space. We shall finally show that the subcategory consisting of modules of G-dimension zero over $R$ is not necessarily a contravariantly finite subcategory in the category of finitely generated $R$-modules.