In order to solve the problems of the short girth and the insufficient error-correction performance for quasi-cyclic low-density parity-check (QC-LDPC) codes, a novel construction method for QC-LDPC codes based on the protograph is proposed. In this method, the protograph whose code-length and code-rate could be flexibly adjusted was selected as the base matrix. Then the Lucas sequence and arithmetic progression with special properties were combined to construct the parity check matrix with ring-length of at least 8 through the low decoding threshold of the protograph and the special properties of the sequence, which required less storage space and was easy to be implemented in hardware. The simulation results show that the PLA-QC-LDPC (2400,1200) code constructed by this method shows a certain degree of net coding gain compared to the fast codable irregular LG-QC-LDPC code based on the Lucas sequence and the greatest common divisor (GCD) sequence, the PM-QC-LDPC code based on the prime number and the multiplication table, and the irregular PL-QC-LDPC code based on the protograph and the eliminating basic trapping sets with the same code-length and code-rate.