Abstract
An approach to improve the performance of QC-LDPC codes is the removal of harmful trapping sets by increasing the girth. However, constructing these LDPC codes with large column weights and girth more than 8 is not easy. We are concerned with protograph-based LDPC codes with large column weights and free of small size trapping sets. We use the edge-coloring technique and some concepts from graph theory such as rainbow cycles to show that for large column weights the removal of all 8-cycles but the ones we call <inline-formula> <tex-math notation="LaTeX">$\textit {rainbow 8-cycle}$ </tex-math></inline-formula> causes the elimination of several small size trapping sets. We provide a detailed theoretical analysis of these harmful trapping sets. Then, we apply them to array-based LDPC codes to significantly simplify and optimize the necessary and sufficient conditions to eliminate those 8-cycles from the Tanner graph. The given exponent matrices and simulation results show the impact of this simplification and the removal of the above mentioned 8-cycles.
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