  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 $\frac{b}{a}<1$, 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.
