Abstract: The construction of non-binary quasi-cyclic low-density parity-check codes are investigated. Important factors that influence the performance of non-binary LDPC codes are discussed in detail. The circle length and connectivity are taken into account in the construction. The selection of shift values to extend the mother matrix and the replacement of non-zero elements over GF(q) for the parity check matrix are studied. Meanwhile, a suboptimum method with lower complexity is proposed to search for the solution according to the new formula. The method is applied to the construction of non-binary LDPC codes over different Galois fields. Simulation results show that the codes constructed by this method outperform the corresponding binary codes by 0.2 dB at FER around10-4 on the BPSK AWGN channel. The performance gap is larger when the order of the Galois field equals the order of the modulation. Compared with non-binary cycle code with the same code rate and approximate code length, the constructed non-binary LDPC code gets an improvement of 0.25 dB at FER around 10-4.