Information Theory and Applications Workshop (ITA)

Quasi-cyclic LDPC codes on Latin squares and the ranks of their parity-check matrices

Li Guo Zhang, Qin Huang, Shu Lin, Khaled Abdel-Ghaffar, Ian F. Blake

Published Jan 1, 2010• 9 citations

Abstract

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.

DOI: 10.1109/ita.2010.5454149

Related papers

Quasi-Cyclic LDPC Codes: An Algebraic Construction, Rank Analysis, and Codes on Latin Squares

Li Guo Zhang, Qin Huang, Shu Lin +2 more

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.

IEEE Transactions on CommunicationsOct 1, 2010120 citations

Quasi-cyclic LDPC codes: Construction and rank analysis of their parity-check matrices

Keke Liu, Qin Huang, Shu Lin +1 more

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.

Information Theory and Applications Workshop (ITA)Feb 1, 201215 citations

A transform approach for computing the ranks of parity-check matrices of quasi-cyclic LDPC codes

Qiuju Diao, Qin Huang, Shu Lin +1 more

Several classes of quasi-cyclic LDPC codes have been proposed in the literature and shown to have excellent performance over noisy channels when decoded with iterative message-passing algorithms. However, by and large, important properties of the codes, including their dimensions, are only given for specific codes based on computer programming. Using Fourier transforms, it is shown that the ranks of parity-check matrices of quasi-cyclic codes can be computed. From these ranks, the dimensions of the codes can be determined. The approach, which unifies most of the known algebraic constructions, is given in detail for three large classes of quasi-cyclic LDPC codes which appear in the literature.

IEEE International Symposium on Information Theory (ISIT)Jul 1, 20118 citations

Quasi-Cyclic LDPC Codes on Cyclic Subgroups of Finite Fields

Li Guo Zhang, Shu Lin, Khaled Abdel-Ghaffar +2 more

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.

IEEE Transactions on CommunicationsJun 28, 201134 citations

Performance Analysis and Design of (3,ρ) — Regular Quasi Cyclic LDPC Codes

Xu Xia, Moon Ho Lee, Choi Seung Je +1 more

In this paper we construct a novel family of (3,rho)-regular quasi-cyclic (QC) low-density parity-check (LDPC) codes, derived from permutation theorem for Latin squares, when used over binary input additive white Gaussian noise (BIAWGN) channel. The codes have high code rate, girth absence of 4, good minimum distance, and sparse parity check matrices. They are designed to perform well when iteratively decoding with sum-product decoding algorithm and to allow low complexity encoding

Dec 10, 20050 citations

Circulant Arrays on Cyclic Subgroups of Finite Fields: Rank Analysis and Construction of Quasi-Cyclic LDPC Codes

Li Guo Zhang, Shu Lin, Khaled Abdel-Ghaffar +2 more

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.

arXiv:1004.1184Apr 7, 20100 citations