On a class of column-weight 3 decomposable LDPC codes with the analysis of elementary trapping sets

A column-weight $k$ LDPC code with the parity-check matrix $H$ is called decomposable if there exists a permutation $\pi$ on the rows of $H$, such that $\pi \left(H\right)$ can be decomposed into $k$ column-weight one matrix. In this paper, some variations of edge coloring of graphs are used to construct some column-weight three decomposable LDPC codes with girths at least six and eight. Applying the presented method on several known classes of bipartite graphs, some classes of column-weight three decomposable LDPC codes are derived having flexibility in length and rate. Interestingly, the constructed parity-check matrices based on the proper edge coloring of graphs can be considered as the base matrix of some high rate column-weight three quasi-cyclic (QC) LDPC codes with maximum-achievable girth 20. The paper also leads to a simple characterization of elementary trapping sets of the decomposable codes based on the chromatic index of the corresponding normal graphs. This characterization corresponds to a simple search algorithm finds all possible existing elementary trapping sets in a girth-6 or girth-8 column-weight 3 LDPC code which are layered super set of a short cycle in the Tanner graph of the code. Simulation results indicate that the QC-LDPC codes with large girths lifted from the constructed base matrices have good performances over AWGN channel.