Aiming at the problem that there may be an error floor of quasi-cyclic low-density parity-check (QC-LDPC) codes in the higher signal-to-noise ratio (SNR) region, a novel construction method of QC-LDPC codes based on the arithmetic sequence (AP) and eliminating elementary trap sets (ETS) is proposed. The method uses the improved ETS elimination algorithm to construct the base matrix and reduce the small ETS, and then applies the AP with the specific character to confirm the cyclic shift coefficient, thus the base matrix is extended to obtain the final check matrix, furthermore, the proposed construction method has the low computing complexity and can flexibly design the code-length and the code-rate. The simulation results show that at the bit error rate of 10-6, the net coding gain of the progressive edge growth(PEG)-trap set-AP(PTAP)-QC-LDPC(1200,600) code with a code-rate of 0.5, compared with those of the QC-LDPC (1200,600) code in IEEE 802.16 standard, the PEG-AP-QC-LDPC(1200,600) code based on PEG algorithm and AP, the CC-QC-LDPC(1200,600)code based on controlling cycles (CC) and the AP-QC-LDPC(1200,600) code based on AP, is improved up to 0.08,0.31,0.57 and 0.64dB respectively, the constructed PTAP-QC-LDPC(1200,600) code can effectively improve the error correction performance in the higher SNR region without the obvious error floor.