QC-LDPC Codes Construction by Concatenating of Circulant Matrices as Block- Columns (original) (raw)
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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
Quasi-cyclic LDPC codes: Construction and rank analysis of their parity-check matrices
Information Theory and Applications, 2012
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
A subtraction based method for the construction of quasi-cyclic LDPC codes of girth eight
2016 International Siberian Conference on Control and Communications (SIBCON), 2016
This article presents a simple, less computational complexity method for constructing exponent matrix () 3, K having girth at least 8 of quasicyclic low-density parity-check (QC-LDPC) codes based on subtraction method. The construction of code deals with the generation of exponent matrix by three formulas. This method is flexible for any block-column length K. The simulations are shown in comparison with some existing appreciable work. The codes with girth 8 are constructed with circulant permutation matrix (CPM) size
Computing Research Repository - CORR, 2010
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 qua...
Quasi-Cyclic LDPC Codes based on
2014
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.
Structured quasi-cyclic LDPC codes with girth 18 and column-weight J>=3
A class of maximum-girth geometrically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes with column-weight J 3 is presented. The method is based on the slope concept between two circulant permutation matrices and the concept of slope matrices. A LDPC code presented by a mv × ml parity-check matrix H , consisting of m × m matrices each of which is either a circulant permutation matrix or a matrix with no nonzero entry, is called a m-circulant vm × lm LDPC code, or just a m-circulant LDPC code. Let D be a (v, J) configuration; that is it has v points, its blocks are of size J , and any two points are contained by at most one block. A m-circulant LDPC code with a mv × ml parity-check matrix H is called a configuration-based code if the set P = {1, 2, ... , v} together with B = {B 1 , B 2 , ... , B l } is a configuration where B i is the subset of P specifying the set of nonzero block positions of the ith block-column of H. Let S = (s i, j) v×v be a matrix over Z m. Under a certain condition, the matrix S is called a m-slope-matrix (m-SM) over a given (v, J) configuration D. To any m-SM S over a (v, J) configuration D, with l blocks, a D-based m-circulant vm × lm LDPC code, referred to as a slope matrix (SM) code, is associated. It is shown that the maximum girth achieved by SM codes over a large class of configurations, including any balanced incomplete block design, is 18. A low-complexity algorithm producing such LDPC codes with girth 6 g 18 is given. As a few examples, a set of SM codes based on the Steiner triple systems STS(9) and STS(13), the 15-points 3 × 5 integer lattice, denoted L(3 × 5), and a 12-points configuration, denoted Aff * (16), obtained from the 16-points affine plane Aff(16) are constructed. These codes have rates at least 0.25, 0.5, 0.4 and 0.37, respectively. From performance perspective, the constructed codes with girth g 14 and length from 34,000 to 92,000 bits and the mentioned rates outperform the random-like LDPC codes of the same lengths and rates, and have a waterfall at about 10 −6 BER and 1.5 dB of Eb/N 0 .
Quasi-Cyclic LDPC Codes for Fast Encoding
IEEE Transactions on Information Theory, 2005
In this correspondence we present a special class of quasi-cyclic low-density parity-check (QC-LDPC) codes, called block-type LDPC (B-LDPC) codes, which have an efficient encoding algorithm due to the simple structure of their parity-check matrices. Since the parity-check matrix of a QC-LDPC code consists of circulant permutation matrices or the zero matrix, the required memory for storing it can be significantly reduced, as compared with randomly constructed LDPC codes. We show that the girth of a QC-LDPC code is upper-bounded by a certain number which is determined by the positions of circulant permutation matrices. The B-LDPC codes are constructed as irregular QC-LDPC codes with parity-check matrices of an almost lower triangular form so that they have an efficient encoding algorithm, good noise threshold, and low error floor. Their encoding complexity is linearly scaled regardless of the size of circulant permutation matrices.
Quasi-Cyclic Low-Density Parity-Check Codes From Circulant Permutation Matrices
IEEE Transactions on Information Theory, 2004
In this correspondence, the construction of low-density parity-check (LDPC) codes from circulant permutation matrices is investigated. It is shown that such codes cannot have a Tanner graph representation with girth larger than 12, and a relatively mild necessary and sufficient condition for the code to have a girth of 6 8 10 or 12 is derived. These results suggest that families of LDPC codes with such girth values are relatively easy to obtain and, consequently, additional parameters such as the minimum distance or the number of redundant check sums should be considered. To this end, a necessary condition for the codes investigated to reach their maximum possible minimum Hamming distance is proposed.
Performance Analysis of Quasi-Cyclic Low Density Parity Check Codes
Low-Density Parity-Check codes are the class of linear block codes, which perform the near Shannon limit performance on data transmission. Here, Quasi Cyclic codes are circulant permutation matrices, for the efficient encoding purpose. In this paper, QC-LDPC Codes have significant performance improvement due to the effective iterative Min-Sum decoding algorithm in terms of Bit Error Rate (BER) versus E b /N o with low and high code rates compared to other existing codes. Soft decision decoding and increased number of iterations of QC-LDPC codes has better performance.
A class of invertible circulant matrices for QC-LDPC codes
2008
This paper presents a new class of easily invertible circulant matrices, defined by exploiting the isomorphism from the ring Mn of n times n circulant matrices over GF(p) to the ring Rn = GF(p)[x]/(xn - 1) of the polynomials modulo (xn - 1). Such class contains matrices free of 4-length cycles that, if sparse, can be included in the parity check matrix of QC-LDPC codes. Bounds for the weight of their inverses are also determined, that are useful for designing sparse generator matrices for these error correcting codes.