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International Conference on Signal Processing and Communication Systems

Algebraic constructions of quasi-cyclic LDPC codes based on generators

Ningbo Zhang, Rui Zhang, Guixia Kang, Yanyan Guo

Published Dec 1, 2015• 0 citations
Publisher

Abstract

This paper puts forward two novel algebraic construction of quasi-cyclic low-density parity-check(QC-LDPC) codes. The two constructions are achieved on multiplicative and additive matrix dispersions of two specified base matrices, which are constructed based on multiplicative and cyclic groups, respectively. In addition, making technique in code construction can also be applied to the two constructions. Simulation results show that codes generated in this paper perform well with iterative decoding over AWGN channel. The codes have advantages over Maykay codes in some aspects of code performances. Meanwhile, a code constructed in this paper converges very fast in iterative decoding, which is an important property in high throughput communication system.

DOI
10.1109/icspcs.2015.7391757

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This paper puts forward two novel algebraic construction of quasi-cyclic low-density parity-check(QC-LDPC) codes. The two constructions are achieved on multiplicative and additive matrix dispersions of two specified base matrices, which are constructed based on multiplicative and cyclic groups, respectively. In addition, making technique in code construction can also be applied to the two constructions. Simulation results show that codes generated in this paper perform well with iterative decoding over AWGN channel. The codes have advantages over Maykay codes in some aspects of code performances. Meanwhile, a code constructed in this paper converges very fast in iterative decoding, which is an important property in high throughput communication system.
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Recently, there have been major developments in utilizing the finite fields to construct Low-density Parity-check (LDPC) codes. In this correspondence, an algebraic approach based on the conjugates of primitive elements over finite fields to construct Quasi-Cyclic (QC) Low-Density Parity-Check codes is presented. Proposed QC-LDPC codes provide an excellent error performance with Belief Propagation (BP) decoding over an Additive White Gaussian Noise (AWGN) channel. Based on numerical results, the performance analysis shows that the proposed QC-LDPC codes perform as well as the randomly constructed Progressive edge growth (PEG) LDPC codes and algebraic QC-LDPC in the lower signal-to-noise ratio (SNR) region but outperform their counterparts in the higher SNR region. Also, the codes constructed are QC in nature, so the encoding can be done with shift register circuits having linear complexity.
A new class of quasi-cyclic LDPC codes whose parity-check matrices are arrays of circulant permutation matrices constructed based on cyclic subgroups of finite fields is presented. This class of codes contains several known classes of algebraic quasi-cyclic LDPC codes as subclasses. Experimental results show that the codes constructed perform very well over the AWGN channel when decoded with iterative decoding based on belief propagation. This class of new QC-LDPC codes contains a subclass of codes which have large minimum distances. Combinatorial expressions for the ranks of the parity-check matrices of a subclass of codes constructed based on fields of characteristic two are given.
In this paper, we concentrate on one algebraic-based quasi-cyclic low-density parity-check (QC-LDPC) code constructed from two subsets of a finite field and generalize it to propose a class of variable-rate QC-LDPC (VR-QC-LDPC) codes, whose parity-check matrices are nested horizontally and have constant number of rows. Thus the proposed codes are significant at least in terms of storage complexity and can be simply implemented. The constructed codes also inherit the original algebraic-based QC-LDPC codes and their exponent matrices can be obtained from two subsets of the given finite field. We hereby analyze the structural properties from the isomorphism perspective, and present some rules to significantly prune the size of search space and determine the non-isomorphic exponent matrices. By distinguishing the smaller quantities of non-isomorphic matrices with cycle property metric, we can easily construct a series of nested exponent matrices with better cycle distributions and obtain the VR-QC-LDPC codes. Numerical results demonstrate that the constructed codes have better iterative decoding performance within a range of code rates and decoding iterations.