Abstract:

Cycle distribution of LDPC(Low-Density Parity-Check) codes affects the codes ^,decoding performance and encoding complexity, however it is commonly NP hard to analyse.We propose the rotationdistance for analysis of QCLDPC（Quasi-Cyclic Low-Density Parity-Check） code based on circulant matrices. The circulant submatrices within the paritycheck matrix are treated as a “matrix node” to simplify theTanner graphs of the codes. Thus cycles of QC-LDPC codes can be found efficiently, and we demonstrate the usefulness of the new method by a simple proof of the known result that 12 is an upper limit of the girth of the QC-LDPC codes we considered. Moreover, the cycle analysis based on the new method also reveals relations between decoding performance and the cycle distribution of the code.

Key words: index termsQC-LDPC codes, circulant matrices, rotation distance amalysis, girth, cycle distribution, matrix tanner graph.