Efficient encoding of cycle codes on graphs with large girths

Low-density parity-check (LDPC) codes of column weight two, also called cycle codes, can be constructed based on connected simple graphs. In this paper, efficiently encodable cycle codes and modified Repeat-Accumulate codes based on Hamiltonian graphs with large girths are devised. Specifically, the Hamiltonian property of the graphs is exploited to obtain efficient encoder structures and a concise description for the Hamiltonian graph named LCF (Lederberg-Coxeter-Frucht) notation is used to reduce the complexity of the encoders, especially the storage requirement. Then, using the Hamiltonian property and the LCF notation, structured cycle codes with large girths and low encoding complexity are devised. Simulation results also reveal that the proposed structured codes exhibit better performance compared with other structured cycle codes.