Necessary conditions for quasi-cyclic LDPC codes to have a given girth

Short cycles in the Tanner graph of a low-density parity-check (LDPC) code may cause a severe performance degradation. In this paper, we investigate the cycle properties of quasi-cyclic LDPC (QC-LDPC) codes. We first analyze a necessary and sufficient condition for a cycle of a given length to exist, by using the sequence representation of a parity-check matrix for a QC-LDPC code. We then derive bounds which are necessary conditions for a QC-LDPC code to have a given girth in terms of its parameters. Our necessary conditions are applicable to any regular or irregular QC-LDPC codes as well as they improve the existing bounds for many classes of regular QC-LDPC codes.