Necessary conditions for avoiding cycles of length 4 or 6 in regular quasi-cyclic LDPC codes

In this paper we investigate the cycle properties of regular quasi-cyclic low-density parity-check (QC-LDPC) codes. Using the sequence representation, we analyze the conditions for short cycles to exist in regular QC-LDPC codes and then give necessary conditions for avoiding short cycles in terms of their parameters. Numerical results show that our bounds on the number of rows in the parity-check matrices for regular QC-LDPC codes without cycles of length 4 or 6 are tighter than any other known bounds.