The representation theory of finite groups has become a standard technique used to construct error correcting quantum codes. The most famous of these types of codes are stabilizer codes and Clifford codes. We introduce slightly 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. With this we give new examples 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 haven’t been studied in the same systematic framework as Clifford codes and stabilizer codes have been studied.

This work was supported by The Norwegian Research Council (Project 345433)