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

• Published in: IET communications Vol. 18; no. 9; pp. 583 - 596
• Main Authors: Raeisi, G., Gholami, M.
• Format: Journal Article
• Language: English
• Published: Wiley 01-06-2024
• Subjects: AWGN channels; graph colouring; parity check codes
• ISSN: 1751-8628; 1751-8636
• DOI: 10.1049/cmu2.12762

A column‐weight k$k$ LDPC code with the parity‐check matrix H$H$ is called decomposable if there exists a permutation π$\pi$ on the rows of H$H$, such that π(H)${\pi }(H)$ can be decomposed into k$k$ 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.
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.