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Journal of Communications

High Rate QC-LDPC Codes with Optimization of Maximum Average Girth

Jianrong Bao, Dan He, Bin Jiang, Xianghong Tang

Published Jan 1, 2016• 5 citations
Publisher
DOI
10.12720/jcm.11.10.936-943

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We present an approach to designing capacity-approaching high-girth low-density parity-check (LDPC) codes that are friendly to hardware implementation, and compatible with some desired input code structure defined using a protograph. The approach is based on a mapping of any class of codes defined using a protograph into a family of hierarchical quasi-cyclic (HQC) LDPC codes. Whereas the parity check matrices of standard quasi-cyclic (QC) LDPC codes are composed of circulant submatrices, those of HQC LDPC codes are composed of a hierarchy of circulant submatrices that are, in turn, constructed from circulant submatrices, and so on, through some number of levels. Next, we present a girth-maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high-girth HQC LDPC code, subject to bounds imposed by the fact that HQC codes are still quasi-cyclic. Finally, we discuss how certain characteristics of a code protograph will lead to inevitable short cycles and show that these short cycles can be eliminated using a “squashing” procedure that results in a high-girth QC LDPC code, although not a hierarchical one. We illustrate our approach with three design examples of QC LDPC codes-two girth-10 codes of rates 1/3 and 0.45 and one girth-8 code of rate 0.7-all of which are obtained from protographs of one-sided spatially coupled codes.
We present an approach to designing capacityapproaching high-girth low-density parity-check (LDPC) codes that are friendly to hardware implementation, and compatible with some desired input code structure defined using a protograph. The approach is based on a mapping of any class of codes defined using a protograph into a family of hierarchical quasicyclic (HQC) LDPC codes. Whereas the parity check matrices of standard quasi-cyclic (QC) LDPC codes are composed of circulant sub-matrices, those of HQC LDPC codes are composed of a hierarchy of circulant sub-matrices that are in turn constructed from circulant sub-matrices, and so on, through some number of levels. Next, we present a girth-maximizing algorithm that optimizes the degrees of freedom within the family of codes to yield a high-girth HQC LDPC code, subject to bounds imposed by the fact that that HQC codes are still quasi-cyclic. Finally, we discuss how certain characteristics of a code protograph will lead to inevitable short cycles, and show that these short cycles can be eliminated using a squashing procedure that results in a high-girth QC LDPC code, although not a hierarchical one. We illustrate our approach with three design examples of QC LDPC codes two girth-10 codes of rates 1/3 and 0.45 and one girth-8 code of rate 0.7 all of which are obtained from protographs of one-sided spatially-coupled codes.
In this paper, we propose the construction of quasi-cyclic (QC) LDPC codes based on the modified progressive edge-growth (PEG) algorithm to achieve the maximum local girth. Although the previously designed QC-LDPC codes based on the PEG algorithm has more flexible code rates than the conventional QC-LDPC code, in the design process, multiple choices of the edges may be chosen. In the proposed algorithm, we aim to maximize the girth property by choosing the suitable edges and thus improve the error correcting performance. Simulation results show that the QC-LDPC codes constructed from the proposed method give higher proportion of high local girths than other methods, particularly, at high code rates. In addition, the proposed codes offer superior bit error rate and block error rate performances to the previous PEG-QC codes over the additive white Gaussian noise (AWGN) channel.