Efficient Search of QC-LDPC Codes with Girths 6 and 8 and Free of Elementary Trapping Sets with Small Size

One of the phenomena that influences significantly the performance of low-density parity-check codes is known as trapping sets. An (a, b) elementary trapping set, or simply an ETS where a is the size and b is the number of degree-one check nodes and b a < 1, causes high decoding failure rate and exert a strong influence on the error floor. In this paper, we provide sufficient conditions for exponent matrices to have fully connected (3, n)-regular QC-LDPC codes with girths 6 and 8 whose Tanner graphs are free of small ETSs. Applying sufficient conditions on the exponent matrix to remove some 8-cycles results in removing all 4-cycles, 6-cycles as well as some small elementary trapping sets. For each girth we obtain a lower bound on the lifting degree and present exponent matrices with column weight three whose corresponding Tanner graph is free of certain ETSs.