Construction of Girth-Twelve Column-Weight QC-LDPC Codes: Implementation and Algorithm

This article presents a methodology for constructing quasi-cyclic low-density parity-check (QC-LDPC) codes with girth-twelve properties and fixed column-weight. These codes are particularly significant in communication systems and storage technologies due to their ability to effectively balance error correction performance with implementation complexity. The core advantage of QC-LDPC codes lies in their structured parity-check matrix architecture, which consists of circulant submatrices obtained through cyclic shifts. This structure facilitates efficient hardware implementation through modular design. The code rate can be flexibly controlled by adjusting the row-weight parameter (k), while the overall code length can be scaled by modifying the submatrix size parameter (m). The critical aspect of this construction is the girth-twelve design, which significantly reduces the negative impact of short cycles on decoding performance. A higher girth value implies fewer pseudo-convergence phenomena, thereby enhancing the accuracy of iterative decoding algorithms such as belief propagation. In implementation, special attention must be paid to designing the shift values of submatrices to avoid the formation of 4-cycles and 6-cycles, while ensuring the final structure satisfies the girth-twelve condition. The algorithm typically involves: 1. Initializing a base matrix with desired column-weight 2. Calculating optimal shift values using combinatorial mathematics 3. Validating cycle properties through graph theory analysis 4. Generating the final parity-check matrix using cyclic permutation matrices Key implementation functions include: - circulant_shift(): Handles cyclic permutation of submatrices - girth_check(): Verifies cycle properties in the Tanner graph - matrix_expansion(): Constructs the full parity-check matrix from base parameters This construction method is particularly suitable for high-reliability transmission scenarios such as deep-space communications or 5G/6G standards, where robust error correction capability is essential. The structured nature of these codes allows for efficient encoding through shift-register-based architectures and simplified decoding implementations.