We have made high-level theoretical study on a new type of noble-gas (Ng) containing anions FNgCC(-). The calculated short Ng-CC bond lengths of 1.13, 1.77, 1.89, and 2.04 Å for Ng=He, Ar, Kr, and Xe, respectively, and the electron density distributions indicated strong covalent interactions between the Ng and CC induced by the polarizing fluoride ion. Except for FHeCC(-), the structures of all other FNgCC(-) were predicted to be linear. The intrinsic stability of the FNgCC(-) was studied by calculating the energies of the three-body dissociation reaction: FNgCC(-) → F(-) + Ng + CC and by calculating the energy barriers of the two-body dissociation reaction: FNgCC(-) → Ng + FCC(-). The results showed that FNgCC(-) (Ng=Ar, Kr, Xe) could be kinetically stable in the gas phase with the three-body dissociation energies of 17, 37, and 64 kcal/mol and two body-dissociation barriers of 22, 31, and 42 kcal/mol, respectively, at the coupled-cluster single double (triple)/aug-cc-pVQZ level of theory. The structures and the stability were also confirmed using the multi-reference CASPT2 calculation. Future experimental identification of the FNgCC(-) anions is expected under cryogenic conditions.