Five quantum register error correction code for higher spin systems

I construct a quantum error correction code (QECC) in higher spin systems using the idea of multiplicative group character. Each N-state quantum particle is encoded as five N-state quantum registers. By doing so, this code can correct any quantum error arising from any one of the five quantum registers. This code generalizes the well-known five qubit perfect code in spin-1/2 systems and is shown to be optimal for higher spin systems. I also report a simple algorithm for encoding. The importance of multiplicative group character in constructing QECCs will be addressed.