Design of non-binary quasi-cyclic LDPC codes by ACE optimization (original) (raw)
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Quasi-cyclic low-density parity-check (QC-LDPC) codes based on protographs are of great interest to code designers because analysis and implementation are facilitated by the protograph structure and the use of circulant permutation matrices for protograph lifting. However, these restrictions impose undesirable fixed upper limits on important code parameters, such as minimum distance and girth. In this paper, we consider an approach to constructing QC-LDPC codes that uses a two-step lifting procedure based on a protograph, and, by following this method instead of the usual one-step procedure, we obtain improved minimum distance and girth properties. We also present two new design rules for constructing good QC-LDPC codes using this two-step lifting procedure, and in each case we obtain a significant increase in minimum distance and achieve a certain guaranteed girth compared to one-step circulant-based liftings. The expected performance improvement is verified by simulation results.
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2008 IEEE Information Theory Workshop, 2008
This paper presents algebraic methods for constructing high performance quasi-cyclic LDPC codes over nonbinary fields. Experimental results show that codes constructed based on these methods perform well over the AWGN channel with iterative decoding using a Fast Fourier Transform based sumproduct algorithm. They achieve significantly large coding gains over Reed-Solomon codes of the same lengths and rates decoded with the hard-decision Berlekamp-Massey algorithm, the algebraic soft-decision Kötter-Vardy algorithm, and the Jiang-Narayanan's adaptive belief propagation algorithm. Due to their quasi-cyclic structure, these LDPC codes can be efficiently encoded using simple shift-registers with linear complexity. They have a great potential to replace Reed-Solomon codes for some applications in communication or storage systems for combating mixed types of noise and interferences.
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IEEE Transactions on Communications, 2009
The construction of finite-length irregular LDPC codes with low error floors is currently an attractive research problem. In particular, for the binary erasure channel (BEC), the problem is to find the elements of selected irregular LDPC code ensembles with the size of their minimum stopping set being maximized. Due to the lack of analytical solutions to this problem, a simple but powerful heuristic design algorithm, the approximate cycle extrinsic message degree (ACE) constrained design algorithm, has recently been proposed. Building upon the ACE metric associated with a cycle in a code graph, we introduce the ACE spectrum of LDPC codes as a useful tool for evaluation of codes from selected irregular LDPC code ensembles. Using the ACE spectrum, we generalize the ACE constrained design algorithm, making it more flexible and efficient. We justify the ACE spectrum approach through examples and simulation results.
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2006
This paper presents a general and three specific algebraic methods for constructing efficiently encodable nonbinary quasi-cyclic LDPC codes. Three classes of quasi-cyclic LDPC codes over nonbinary finite fields are constructed. Codes constructed perform very well over the AWGN channel with iterative decoding and achieve large coding gains over the Reed-Solomon codes of the same parameters. Nonbinary LDPC codes may be used to replace Reed-Solomon codes in some communication environments or storage systems for combating mixed types of noises and interferences.
Quasi-cyclic generalized ldpc codes with low error floors
IEEE Transactions on Communications, 2008
In this paper, a novel methodology for designing structured generalized LDPC (G-LDPC) codes is presented. The proposed design results in quasi-cyclic G-LDPC codes for which efficient encoding is feasible through shift-register-based circuits. The structure imposed on the bipartite graphs, together with the choice of simple component codes, leads to a class of codes suitable for fast iterative decoding. A pragmatic approach to the construction of G-LDPC codes is proposed. The approach is based on the substitution of check nodes in the protograph of a low-density parity-check code with stronger nodes based, for instance, on Hamming codes. Such a design approach, which we call LDPC code doping, leads to low-rate quasi-cyclic G-LDPC codes with excellent performance in both the error floor and waterfall regions on the additive white Gaussian noise channel.
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2014 Information Theory and Applications Workshop (ITA), 2014
Presented in this paper is a reduced-complexity iterative decoding scheme for quasi-cyclic (QC) LDPC codes. This decoding scheme is devised based on the section-wise cyclic structure of the parity-check matrix of a QC-LDPC code. Using this decoding scheme, the hardware implementation complexity of a QC-LDPC decoder can be significantly reduced without performance degradation. A high-rate QC-LDPC code that can achieve a very low error-rate without a visible error-floor is used to demonstrate the effectiveness of the proposed decoding scheme. Also presented in this paper are two other high-rate QC-LDPC codes and a method for constructing rate-1 2 QC-LDPC codes whose Tanner graphs have girth 8. All the codes constructed perform well with low error-floor using the proposed decoding scheme.
A Matrix-Theoretic Approach to the Construction of Non-Binary Quasi-Cyclic LDPC Codes
IEEE Transactions on Communications, 2015
This paper presents two simple and very flexible methods for constructing non-binary (NB) quasi-cyclic (QC) LDPC codes. The proposed construction methods have several known ingredients including base array, masking, binary to nonbinary replacement and matrix-dispersion. By proper choice and combination of these ingredients, NB-QC-LDPC codes with excellent performance can be constructed. The constructed codes can be decoded with a reduced-complexity iterative decoding scheme which significantly reduces the hardware implementation complexity. I. INTRODUCTION L DPC CODES, discovered in 1962 [1] and rediscovered in late 1990's [2], [3], are currently the most promising coding technique for error control in communication and data storage systems due to their capacity-approaching performances and practically implementable decoding algorithms. Since their rediscovery, a great deal of research effort has been expended in design, analysis, decoding, generalizations and applications of these amazing codes. However, most of the research effort has been focused only on binary LDPC codes. Research effort expended in non-binary (NB) LDPC codes is far less than that devoted to their binary counterparts. This lack of enthusiasm in NB-LDPC codes may be due to the concern of their decoding complexity in both computation and hardware implementation. NB-LDPC codes do have advantages over their binary counterparts for communication and data storage channels where both random and burst errors occur simultaneously. Furthermore, for using high-order modulations with large signal constellations for communication, it is very natural to use NB-LDPC codes. For all of these reasons, NB-LDPC codes deserve more attention and research effort. There are various types of LDPC codes. Among them, the most preferred type of LDPC codes for practical applications in communication and storage systems are LDPC codes with quasi-cyclic (QC) structure, called QC-LDPC codes [4], [5]. A QC-LDPC code is given by the null space of an array H of sparse circulant matrices of the same size over a finite field, binary or non-binary. In most of the constructions of QC-LDPC codes, the sparse circulant matrices in the paritycheck array H of a QC-LDPC code are circulant permutation matrices (CPMs). Such a parity-check array H of a QC-LDPC
IEEE Transactions on Information Theory, 2012
Quasi-cyclic (QC) low-density parity-check (LDPC) codes are an important instance of proto-graph-based LDPC codes. In this paper we present upper bounds on the minimum Hamming distance of QC LDPC codes and study how these upper bounds depend on graph structure parameters (like variable degrees, check node degrees, girth) of the Tanner graph and of the underlying proto-graph. Moreover, for several classes of proto-graphs we present explicit QC LDPC code constructions that achieve (or come close to) the respective minimum Hamming distance upper bounds. Because of the tight algebraic connection between QC codes and convolutional codes, we can state similar results for the free Hamming distance of convolutional codes. In fact, some QC code statements are established by first proving the corresponding convolutional code statements and then using a result by Tanner that says that the minimum Hamming distance of a QC code is upper bounded by the free Hamming distance of the convolutional code that is obtained by "unwrapping" the QC code.
Constructions of Nonbinary Quasi-Cyclic LDPC Codes: A Finite Field Approach
In the late 1950's and early 1960's, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for correcting random errors with algebraic decoding, such as Bose-Chaudhuri-Hocqenghem (BCH) and Reed-Solomon (RS) codes. Recently it has been shown that finite fields can also be used successfully to construct binary quasi-cyclic (QC)-LDPC codes that perform very well not only over the AWGN channel but also over the binary erasure channel with iterative decoding, besides being efficiently encodable. This paper is concerned with constructions of nonbinary QC-LDPC codes based on finite fields.
Quasi-Cyclic LDPC Codes of Column-Weight Two Using a Search Algorithm
EURASIP Journal on Advances in Signal Processing, 2007
This article introduces a search algorithm for constructing quasi-cyclic LDPC codes of column-weight two. To obtain a submatrix structure, rows are divided into groups of equal sizes. Rows in a group are connected in their numerical order to obtain a cyclic structure. Two rows forming a column must be at a specified distance from each other to obtain a given girth. The search for rows satisfying the distance is done sequentially or randomly. Using the proposed algorithm regular and irregular column-weight-two codes are obtained over a wide range of girths, rates, and lengths. The algorithm, which has a complexity linear with respect to the number of rows, provides an easy and fast way to construct quasi-cyclic LDPC codes. Constructed codes show good bit-error rate performance with randomly shifted codes performing better than sequentially shifted ones.