The Quantum Hamming and Hexacodes

In this paper we present the relations between algebra and physics in the context of quantum error-correcting codes, showing the connection between Pauli matrices, finite fields, classical and quantum error-correcting codes. We show how to encode and especially decode (i.e., correct errors) quantum error-correcting codes. This is exemplarily demonstrated for the cyclic quantum Hamming code of length five and its extension, the quantum Hexacode.