Quantum error correcting codes: From stabilizer codes to induced codes

The first large class of quantum error-correcting codes that was constructed is the class of stabilizer codes. The construction method used for these codes is based on finding abelian subgroups of general Pauli groups. Stabilizer codes are a special case of a more general class of quantum error-correcting codes---Clifford codes. The construction technique for Clifford codes generalizes the stabilizer code construction by introducing abstract error groups. Abstract error groups generalize the Pauli groups so that codes can be constructed for quantum systems with any dimension. Clifford codes are thus constructed by finding normal subgroups of abstract error groups. In this dissertation we show that Clifford codes are a special case of a larger class of codes, which we refer to as induced codes. In addition to this new construction method, we also present some new results pertaining to a subclass of stabilizer codes called degenerate stabilizer codes.