A low complexity construction method of quasi-cyclic low-density parity check(QC-LDPC) code with the characteristic of arithmetic progression(AP) was proposed. The construction method has simple structure, thus it can save storage space, and regulate the code-length and code-rate flexibly according to the actual demand. A novel AP-QC-LDPC(4599,4307) code is constructed by applying the proposed method, each row elements of the parity check matrix constitutes arithmetic progressions with the monotonically increasing tolerances, so the check matrix does not contain girth-4. The simulation analysis shows that at the bit error rate(BER) of 10-6, the net coding gain(NCG) of the constructed AP-QC-LDPC(4599,4307) code is 2.19dB higher than that of RS(255,239) code in ITU-T G.975, and 1.48dB higher than that of LDPC(32640,30592) code in ITU-T G.975.1. In addition, the NCGs of AP-QC-LDPC(4599,4307) code are respectively 0.16dB and 0.2dB higher than that of both eIRA-QC-LDPC(4599,4307) code and QC-LDPC(3780,3540) code based on finite field multiplicative group. The constructed AP-QC-LDPC(4599,4307) code with better error-correction performance is more suitable for the requirements of optical communication systems.