Objective: Quasi-cyclic low-density parity-check (QC-LDPC) codes are a class of channel codes known for their quasi-cyclic structure and sparse parity-check matrices. They have attracted significant attention due to their relatively simple hardware implementation and superior decoding performance. However, short cycles—particularly four and six cycles—can severely degrade the decoding performance of QC-LDPC codes. A key challenge in current LDPC code construction research is designing QC-LDPC codes with flexible circulant sizes and column weights while eliminating four and six cycles. Existing methods to eliminate short cycles can be broadly divided into two categories: search-based and explicit methods. Search-based methods provide parameter flexibility but involve high descriptive complexity. In contrast, explicit methods construct QC-LDPC codes directly through formulas, resulting in low descriptive complexity. In recent years, several explicit methods based on combinatorial mathematics have been introduced, with disjoint difference sets (DDS) playing a central role in constructing QC-LDPC codes free of four and six cycles. However, existing methods cannot simultaneously achieve the elimination of four and six cycles, support diverse column weights, and offer flexible circulant sizes. To address this gap, this study proposes an explicit construction method for QC-LDPC codes based on DDS that satisfies all three requirements. Methods: This study integrates DDS with row-weight extension (RE) techniques to propose a new construction method termed DDS-RE. First, the existence of four and six cycles in the Tanner graph corresponding to the new construction is analyzed using cycle control equations. Second, the flexibility of circulant size selection in the new construction is compared with the original DDS method. Finally, bit error rate (BER) and block error rate (BLER) performances of the newly constructed QC-LDPC codes are evaluated through simulations using the sum-product algorithm (SPA) and compared with several representative explicit methods: the greatest-common-divisor (GCD), Golomb ruler (GR), Sidon sequence (SS), and the original DDS methods. Additionally, decoding performance is compared with two construction methods incorporating RE techniques (GCD-RE and SS-RE). Results: This study rigorously proves, using cycle control equations, that the Tanner graph of the QC-LDPC codes constructed by the new method is free of four and six cycles. Analysis reveals that, compared with existing DDS methods, the newly proposed DDS-RE method enables smaller circulant sizes, particularly for larger column weights, where circulant size flexibility is significantly enhanced. Simulation results of decoding performance demonstrate that the new codes outperform GR codes in the high signal-to-noise ratio (SNR) region while also offering flexible column weights and circulant sizes. They also perform comparably to DDS-based codes but with the added advantage of circulant size flexibility. Furthermore, the DDS-RE codes significantly outperform those constructed by the GCD and GCD-RE methods. In high SNR regions, they also slightly surpass SS-RE codes while benefiting from flexible circulant sizes. Conclusions: By combining the DDS method with RE techniques, this study presents a new explicit construction method for QC-LDPC codes—DDS-RE. The resulting codes are guaranteed to be free of four and six cycles and support flexible circulant sizes. Compared with existing explicit methods and other RE-based methods, DDS-RE constructs codes with excellent performance while simultaneously achieving the elimination of four and six cycles, diverse column weights, and flexible circulant sizes. This study offers new insights into the explicit construction of high-performance QC-LDPC codes.