On the Structure of Nonstabilizer Clifford Codes

Clifford codes are a class of quantum error control codes that form a natural generalization of stabilizer codes, These codes were introduced in 1996 by Knill, but only a single Clifford code was known, which was not already a stabilizer code. We derive a necessary and sufficient condition that allows one to decide when a Clifford code is a stabilizer code, and compile a table of all true Clifford codes for error groups of small order.