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IEEE International Symposium on Information Theory (ISIT)

Protograph design with multiple edges for regular QC LDPC codes having large girth

Hosung Park, Seokbeom Hong, Jong‐Seon No, Dong‐Joon Shin

Published Jul 1, 2011• 7 citations
Publisher

Abstract

In this paper, all subgraph patterns of protographs which prevent quasi-cyclic (QC) low-density parity-check (LDPC) codes from having large girth are searched in allowance with multiple edges based on graph theoretic approach. A systematic construction of protographs with multiple edges using combinatorial design is proposed for designing QC LDPC codes with girth greater than or equal to 14.

DOI
10.1109/isit.2011.6034272

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IEEE Transactions on Communications•Jun 9, 2020•28 citations
In this paper, all subgraph patterns of protographs which prevent quasi-cyclic (QC) low-density parity-check (LDPC) codes from having large girth are searched in allowance with multiple edges based on graph theoretic approach. A systematic construction of protographs with multiple edges using combinatorial design is proposed for designing QC LDPC codes with girth greater than or equal to 14.
There have been lots of efforts on the construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes with large girth. However, most of them focus on protographs with single edges and little research has been done for the construction of QC LDPC codes lifted from protographs with multiple (i.e., parallel) edges. Compared to single-edge protographs, multiple-edge protographs have benefits such that QC LDPC codes lifted from them can potentially have larger minimum Hamming distance. In this paper, all subgraph patterns of multiple-edge protographs, which prevent QC LDPC codes from having large girth by inducing inevitable cycles, are fully investigated based on a graph-theoretic approach. By using combinatorial designs, a systematic construction method of multiple-edge protographs is proposed for regular QC LDPC codes with girth at least 12 and another method is proposed for regular QC LDPC codes with girth at least 14. Moreover, a construction algorithm of QC LDPC codes based on certain liftings of multiple-edge protographs is proposed and it is shown that the resulting QC LDPC codes have larger upper bounds on the minimum Hamming distance than those lifted from single-edge protographs. Simulation results are provided to compare the performance of the proposed QC LDPC codes with progressive edge-growth (PEG) LDPC codes and with PEG QC LDPC codes.
There have been lots of efforts on the construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes with large girth. However, most of them are focused on protographs with single edges and little research has been done for the construction of QC LDPC codes lifted from protographs with multiple edges. Compared to single-edge protographs, multiple-edge protographs have benefits such that QC LDPC codes lifted from them can potentially have larger minimum Hamming distance. In this paper, all subgraph patterns of multiple-edge protographs, which prevent QC LDPC codes from having large girth by inducing inevitable cycles, are fully investigated based on graph-theoretic approach. By using combinatorial designs, a systematic construction method of multiple-edge protographs is proposed for regular QC LDPC codes with girth at least 12 and also other method is proposed for regular QC LDPC codes with girth at least 14. A construction algorithm of QC LDPC codes by lifting multiple-edge protographs is proposed and it is shown that the resulting QC LDPC codes have larger upper bounds on the minimum Hamming distance than those lifted from single-edge protographs. Simulation results are provided to compare the performance of the proposed QC LDPC codes, the progressive edge-growth (PEG) LDPC codes, and the PEG QC LDPC codes.
A quasi-cyclic (QC) low-density parity-check (LDPC) code can be viewed as the protograph code with circulant permutation matrices. In this paper, we find all the subgraph patterns of protographs of QC LDPC codes having inevitable cycles of length 2i,i= 6,7,8,9,10, i.e., the cycles existing regardless of the shift values of circulants. It is also derived that if the girth of the protograph is 2g,gges 2, its protograph code cannot have the inevitable cycles of length smaller than 6g. Based on these subgraph patterns, we propose new combinatorial construction methods of the protographs, whose protograph codes can have girth larger than or equal to 14.
A quasi-cyclic (QC) low-density parity-check (LDPC) code can be viewed as the protograph code with circulant permutation matrices (or circulants). In this correspondence, we find all the subgraph patterns of protographs of QC LDPC codes having inevitable cycles of length 2i, i = 6, 7, 8, 9,10, i.e., the cycles that always exist regardless of the shift values of circulants. It is also derived that if the girth of the protograph is 2g, g > 2, its protograph code cannot have the inevitable cycles of length smaller than 6g. Based on these subgraph patterns, we propose new combinatorial construction methods of the protographs, whose protograph codes can have girth larger than or equal to 14 or 18. We also propose a couple of shift value assigning rules for circulants of a QC LDPC code guaranteeing the girth 14.
Spatially coupled (SC) low-density parity-check (LDPC) codes can achieve capacity approaching performance with low message recovery latency when using sliding window (SW) decoding. An SC-LDPC code constructed from a protograph can be generated by first coupling a chain of block protographs and then lifting the coupled protograph using permutation matrices. In this paper, we introduce a systematic design to eliminate 4-cycles in a coupled protograph. Further using a quasi-cyclic (QC) lifting, we introduce a procedure for constructing QC-SC-LDPC codes of girth at least eight. This can be interpreted as a multi-stage graph lifting process that yields a greater flexibility in designing QC-SC-LDPC codes with a large girth than previous approaches. Simulation results show the design leads to improved decoding performance, particularly in the error floor, compared to random constructions. Finally, we determine the minimum coupling width required to eliminate 4-cycles in a coupled protograph.