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IEEE Transactions on Communications

Quasi-Cyclic LDPC Codes on Cyclic Subgroups of Finite Fields

Li Guo Zhang, Shu Lin, Khaled Abdel-Ghaffar, Zhi Jun Ding, Bo Zhou

Published Jun 28, 2011• 34 citations
Publisher

Abstract

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.

DOI
10.1109/tcomm.2011.060911.100208

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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.
This paper consists of three parts. The first part presents a large class of new binary quasi-cyclic (QC)-LDPC codes with girth of at least 6 whose parity-check matrices are constructed based on cyclic subgroups of finite fields. Experimental results show that the codes constructed perform well over the binary-input AWGN channel with iterative decoding using the sum-product algorithm (SPA). The second part analyzes the ranks of the parity-check matrices of codes constructed based on finite fields with characteristic of 2 and gives combinatorial expressions for these ranks. The third part identifies a subclass of constructed QC-LDPC codes that have large minimum distances. Decoding of codes in this subclass with the SPA converges very fast.
This paper consists of three parts. The first part presents a large class of new binary quasi-cyclic (QC)-LDPC codes with girth of at least 6 whose parity-check matrices are constructed based on cyclic subgroups of finite fields. Experimental results show that the codes constructed perform well over the binary-input AWGN channel with iterative decoding using the sum-product algorithm (SPA) and they outperform the corresponding pseudo-random QC-LDPC codes constructed with the PEG-algorithm. The second part analyzes the ranks of the parity-check matrices of codes constructed based on finite fields with characteristic of 2 and gives combinatorial expressions for these ranks. The third part identifies a subclass of constructed QC-LDPC codes that have large minimum distances. Decoding of codes in this subclass with the SPA converges very fast.
A construction of binary and non-binary quasi-cyclic (QC)-LDPC codes based on partitions of finite fields of characteristic 2 is proposed. The construction is carried out in the Fourier transform domain. The parity-check matrices of these QC-LDPC codes are arrays of circulant permutation matrices. The ranks of these arrays are analyzed and combinatorial expressions are derived. Example codes are given and simulations show that they perform well over the AWGN channel decoded with message-passing decoding algorithms.
Quasi-cyclic LDPC codes are the most promising class of structured LDPC codes due to their ease of implementation and excellent performance over noisy channels when decoded with message-passing algorithms as extensive simulation studies have shown. In this paper, an approach for constructing quasi-cyclic LDPC codes based on Latin squares over finite fields is presented. By analyzing the parity-check matrices of these codes, combinatorial expressions for their ranks and dimensions are derived. Experimental results show that, with iterative decoding algorithms, the constructed codes perform very well over the AWGN and the binary erasure channels.
On dispersing different base matrices combined with masking, different kinds of quasi-cyclic low-density parity check (QC-LDPC) codes are constructed, which may achieve good error performances. Two new algebraic constructions of regular QC-LDPC codes are presented in this paper. The two constructions are based on two base matrices by dispersing and masking. The two base matrices are based on the generators of the cyclic group of a prime field. Results show that the two classes of codes presented in this paper perform well with iterative decoding over AWGN channel and have advantages over MacKay codes in some aspects of code performances. We find that one of the constructed codes converges very fast with iterative decoding, which is a critical property in ultra-high throughput communication systems.
Quasi-cyclic codes are the most promising class of structured LDPC codes due to their ease of implementation and excellent performance over noisy channels when decoded with message-passing algorithms as extensive simulation studies have shown. An approach for constructing quasi-cyclic LDPC codes based on Latin squares over finite fields is presented. By analyzing the parity-check matrices of these codes, expressions for their ranks are derived. Experimental results show that, with iterative decoding algorithms, the constructed codes perform very well over the AWGN and the binary erasure channels.