Three MacWilliams type identities and quantum error-correcting <inline-formula><tex-math id="M1">$ G $</tex-math></inline-formula>-codes

We prove three MacWilliams type identities for irreducible projective representations of finite groups. As an application, we deduce the MacWilliams identities of weight enumerators, double weight enumerators and complete weight enumerators for quantum error-correcting $ G $-codes and obtain the Singleton bounds.