To improve the efficiency of repairing faulty nodes in a distributed storage system, this study proposes a new construction algorithm for fractional repetition (FR) codes. The algorithm resorts to the factorization of the complete graph for node construction, and the nodes constructed are referred to as complete graph factorization based FR (CGFBFR) nodes. Specifically, the complete graph is factorized, and the number of factors generated from the complete graph after the factorization is completed is determined. The number of factors of the complete graph is selected according to the repetition degrees of the data blocks that need to be stored. All the vertices of the selected factors of the complete graph are regarded as the data blocks that need to be stored in the distributed storage system. Then, the edges of the selected factor graph are marked and stored as distributed data nodes. Finally, an encoding matrix is generated with the vertices and edges of the selected factors, and the distributed storage system stores the data blocks respectively according to the data in the encoding matrix. The experimental simulation results show that compared with the Reed-Solomon (RS) codes, the simple regenerating codes (SRCs), and the latest cyclic variable FR (VFR) codes in the distributed storage system, the codes generated by the new FR code construction algorithm proposed in this study can quickly repair the faulty node when the system repairs the node. The proposed algorithm can be widely applied to distributed storage systems as it reduces the repair bandwidth overhead, repair locality, and repair complexity of the faulty node, provides a simple construction process, and allows flexible selection of construction parameters.