The Essential Conditions of Quasi-cyclic LDPC Codes without Girth 4

The essential conditions for quasi-cyclic low-density parity-check codes(QC-LDPC codes) to have no Girth 4 are given.This solves the problem of designing without-girth-4 QC-LDPC codes of any length.Adjusting the dimension and shift factors of the circulant sub-matrices of the given sparse parity-check matrices according to the proposed theorems,the QC-LDPC codes without girth 4 are constructed.The matrice equation set for girth 4 check is established.Through setting the dimension and shift factors of the circulant sub-matrices,we can check whether the QC-LDPC codes have Girth 4 or not.We can multiply the dimension of the circulant sub-matrices to get the QC-LDPC codes without Girth 4 of different lengths.Our theorems keep the QC-LDPC codes without Girth 4 along with increasing of the code length.Compared to the previous algorithms,the proposed test algorithm decreases computation,shortens the test time and reduces the complexity of structuring the parity-check matrix.Experimental results indicate that the designed QC-LDPC codes achieve good bit error rate(BER) performance.