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Information Theory and Applications Workshop (ITA)

A transform approach for constructing quasi-cyclic Euclidean geometry LDPC codes

Qiuju Diao, Wei Gang Zhou, Shu Hwa Lin, Khaled Abdel-Ghaffar

Published Feb 1, 2012• 3 citations
Publisher

Abstract

A method for constructing quasi-cyclic Euclidean geometry (QC-EG) LDPC codes in the Fourier transform domain is presented. Given a Euclidean geometry over a finite field of characteristic 2, base matrices in the Fourier transform domain are first constructed. Then the inverse Fourier transforms of these base matrices, combined with row and column permutations, result in low-density arrays of circulant permutation matrices and/or zero matrices. The null spaces of these low-density arrays give a family of QC-EG-LDPC codes. Codes in a special subclass have large minimum distances and their Tanner graphs contain no harmful trapping sets with sizes smaller than their minimum distances.

DOI
10.1109/ita.2012.6181800

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IEICE Transactions on Communications•Jan 1, 2012•0 citations
A method for constructing quasi-cyclic Euclidean geometry (QC-EG) LDPC codes in the Fourier transform domain is presented. Given a Euclidean geometry over a finite field of characteristic 2, base matrices in the Fourier transform domain are first constructed. Then the inverse Fourier transforms of these base matrices, combined with row and column permutations, result in low-density arrays of circulant permutation matrices and/or zero matrices. The null spaces of these low-density arrays give a family of QC-EG-LDPC codes. Codes in a special subclass have large minimum distances and their Tanner graphs contain no harmful trapping sets with sizes smaller than their minimum distances.
An approach for studying quasi-cyclic codes based on matrix transformations via Fourier transforms and row and column permutations is presented. These transformations put a parity-check matrix in the form of an array of circulant matrices into a diagonal array of matrices of the same size over an extension field. The approach is used to characterize certain structural properties of low-density parity-check (LDPC) codes such as the girths of their Tanner graphs. Many constructions of quasi-cyclic LDPC codes can be unified under the proposed approach.
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 is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly low-density parity-check (LDPC) codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterization of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite-geometry LDPC codes using the results developed in the first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping set structure of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite-geometry and finite-field cyclic and quasi-cyclic LDPC codes with large minimum distances are shown to have no harmful trapping sets of size smaller than their minimum distances. Consequently, their error-floor performances are dominated by their minimum distances.
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.
This paper is concerned with construction and structural analysis of both cyclic and quasi-cyclic codes, particularly LDPC codes. It consists of three parts. The first part shows that a cyclic code given by a parity-check matrix in circulant form can be decomposed into descendant cyclic and quasi-cyclic codes of various lengths and rates. Some fundamental structural properties of these descendant codes are developed, including the characterizations of the roots of the generator polynomial of a cyclic descendant code. The second part of the paper shows that cyclic and quasi-cyclic descendant LDPC codes can be derived from cyclic finite geometry LDPC codes using the results developed in first part of the paper. This enlarges the repertoire of cyclic LDPC codes. The third part of the paper analyzes the trapping sets of regular LDPC codes whose parity-check matrices satisfy a certain constraint on their rows and columns. Several classes of finite geometry and finite field cyclic and quasi-cyclic LDPC codes with large minimum weights are shown to have no harmful trapping sets with size smaller than their minimum weights. Consequently, their performance error-floors are dominated by their minimum weights.
Quasi-cyclic (QC) low-density parity-check (LDPC) codes have several appealing properties regarding decoding, storage requirements and encoding aspects. In this paper, we focus on the QC LDPC codes over GF(q) whose parity-check matrices have fixed column weight j=2. By investigating two subgraphs in the Tanner graphs of the corresponding base matrices, we derive two upper bounds on the minimum Hamming distance for this class of codes. In addition, a method is proposed to construct QC LDPC codes over GF(q), which have good Hamming distance distributions. Simulations show that our designed codes have good performance.