Short nonbinary maximum distance separable cycle codes

This letter presents a class of very short nonbinary cycle codes that are maximum distance separable (MDS). It is proved that there is one and only one regular graph on which MDS cycle codes may be constructed and only for finite field orders larger than or equal to 5. An explicit construction method is described to generate MDS cycle codes based on the identified graph, for any admissible field order. The proposed codes admit efficient soft-decision decoding based on belief propagation, with small performance losses with respect to optimum maximum-likelihood decoding. When concatenated with an inner binary code, they yield short binary codes with low code rates representing a practical and effective solution for the protection of short messages, for example, critical packet headers in wireless communication systems.