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

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