Projective error models: Stabilizer codes, Clifford codes, and weak stabilizer codes

We introduce more general notions of Clifford codes and stabilizer codes, the latter we call weak stabilizer codes. This is all formulated in the language of projective representation theory of finite groups and we give a novel description of the detectable errors for a Clifford code. We give a complete characterization of when a Clifford code is also a weak stabilizer code in the case where the considered error model is a nice error basis. We also give examples of infinite families of non-stabilizer Clifford codes as well as examples of non-Clifford weak stabilizer codes. The latter of these types of examples is a class of codes that have not been studied in the same systematic framework as Clifford codes and stabilizer codes.