Application of efficient Chase algorithm in decoding of generalized low-density parity-check codes

We consider the iterative decoding of generalized low-density (GLD) parity-check codes where, rather than employ an optimal subcode decoder, a Chase (1972) algorithm decoder more commonly associated with "turbo product codes" is used. GLD codes are low-density graph codes in which the constraint nodes are other than single parity-checks. For extended Hamming-based GLD codes, we use bit error rates derived by simulation to demonstrate this new strategy to be successful at higher code rates. For long block lengths, good performance close to capacity is possible with decoding costs reduced further since the Chase decoder employed is an efficient implementation.     

MacNeish-Mann theorem based iteratively decodable codes for optical communication systems

A novel family of low-density parity-check codes is proposed based on MacNeish-Mann theorem for construction of mutually orthogonal Latin squares. Codes from this family have high code rate, girth at least six, large minimum distance, and significantly outperform conventional forward error correction schemes based on Reed-Solomon (RS) and turbo codes.     

Iteratively decodable codes from orthogonal arrays for optical communication systems

A novel family of low-density parity-check codes is proposed based on orthogonal arrays. Codes from this family have high code rate, girth of at least six, large minimum distance, and significantly outperform the error correction schemes based on turbo product codes proposed for optical communication systems.     

Upper bounds on the rate of LDPC codes as a function of minimum distance

New upper bounds on the rate of low-density parity-check (LDPC) codes as a function of the minimum distance of the code are derived. The bounds apply to regular LDPC codes, and sometimes also to right-regular LDPC codes. Their derivation is based on combinatorial arguments and linear programming. The new bounds improve upon the previous bounds due to Burshtein et al. It is proved that at least for high rates, regular LDPC codes with full-rank parity-check matrices have worse relative minimum distance than the one guaranteed by the Gilbert-Varshamov bound.     

Capacity-approaching protograph codes

This paper discusses construction of protographbased low-density parity-check (LDPC) codes. Emphasis is placed on protograph ensembles whose typical minimum distance grows linearly with block size. Asymptotic performance analysis for both weight enumeration and iterative decoding threshold determination is provided and applied to a series of code constructions. Construction techniques that yield both low thresholds and linear minimum distance growth are introduced by way of example throughout. The paper also examines implementation strategies for high throughput decoding derived from first principles of belief propagation on bipartite graphs.     

LDPC block and convolutional codes based on circulant matrices

A class of algebraically structured quasi-cyclic (QC) low-density parity-check (LDPC) codes and their convolutional counterparts is presented. The QC codes are described by sparse parity-check matrices comprised of blocks of circulant matrices. The sparse parity-check representation allows for practical graph-based iterative message-passing decoding. Based on the algebraic structure, bounds on the girth and minimum distance of the codes are found, and several possible encoding techniques are described. The performance of the QC LDPC block codes compares favorably with that of randomly constructed LDPC codes for short to moderate block lengths. The performance of the LDPC convolutional codes is superior to that of the QC codes on which they are based; this performance is the limiting performance obtained by increasing the circulant size of the base QC code. Finally, a continuous decoding procedure for the LDPC convolutional codes is described.     

Upper bounds on the rate of LDPC codes

We derive upper bounds on the rate of low-density parity-check (LDPC) codes for which reliable communication is achievable. We first generalize Gallager's (1963) bound to a general binary-input symmetric-output channel. We then proceed to derive tighter bounds. We also derive upper bounds on the rate as a function of the minimum distance of the code. We consider both individual codes and ensembles of codes.     

Quasicyclic low-density parity-check codes from circulant permutation matrices

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.     

Several properties of short LDPC codes

In this paper, we present several properties on minimum distance(d/sub min/) and girth(G/sub min/) in Tanner graphs for low-density parity-check (LDPC) codes with small left degrees. We show that the distance growth of (2, 4) LDPC codes is too slow to achieve the desired performance. We further give a tight upper bound on the maximum possible girth. The numerical results show that codes with large G/sub min/ could outperform the average performance of regular ensembles of the LDPC codes over binary symmetric channels. The same codes perform about 1.5 dB away from the sphere-packing bound on additive white Gaussian noise channels.     

On the active distances of partial unit memory codes based on LDPC codes

Partial unit memory (PUM) convolutional codes constructed from block codes make it possible to obtain convolutional codes with high correcting properties. An ensemble of binary PUM convolutional codes based on low-density parity-check block codes is introduced. This ensemble is specified by semi-infinite parity-check matrices with partial unit memory. The asymptotic bounds on a free distance and active row distances are derived for codes of the specified ensemble. It is proved that the ensemble codes have the positive slope of active row distances.     

Performance of low-density parity-check codes with linear minimum distance

This correspondence studies the performance of the iterative decoding of low-density parity-check (LDPC) code ensembles that have linear typical minimum distance and stopping set size. We first obtain a lower bound on the achievable rates of these ensembles over memoryless binary-input output-symmetric channels. We improve this bound for the binary erasure channel. We also introduce a method to construct the codes meeting the lower bound for the binary erasure channel. Then, we give upper bounds on the rate of LDPC codes with linear minimum distance when their right degree distribution is fixed. We compare these bounds to the previously derived upper bounds on the rate when there is no restriction on the code ensemble.     

High-Rate Quasi-Cyclic Low-Density Parity-Check Codes Derived From Finite Affine Planes

This paper shows that several attractive classes of quasi-cyclic (QC) low-density parity-check (LDPC) codes can be obtained from affine planes over finite fields. One class of these consists of duals of one-generator QC codes. Presented here for codes contained in this class are the exact minimum distance and a lower bound on the multiplicity of the minimum-weight codewords. Further, it is shown that the minimum Hamming distance of a code in this class is equal to its minimum additive white Gaussian noise (AWGN) pseudoweight. Also discussed is a class consisting of codes from circulant permutation matrices, and an explicit formula for the rank of the parity-check matrix is presented for these codes. Additionally, it is shown that each of these codes can be identified with a code constructed from a constacyclic maximum distance separable code of dimension 2. The construction is similar to the derivation of Reed-Solomon (RS)-based LDPC codes presented by Chen and Djurdjevic Experimental results show that a number of high rate QC-LDPC codes with excellent error performance are contained in these classes     

LDPC codes from generalized polygons

We use the theory of finite classical generalized polygons to derive and study low-density parity-check (LDPC) codes. The Tanner graph of a generalized polygon LDPC code is highly symmetric, inherits the diameter size of the parent generalized polygon, and has minimum (one half) diameter-to-girth ratio. We show formally that when the diameter is four or six or eight, all codewords have even Hamming weight. When the generalized polygon has in addition an equal number of points and lines, we see that the nonregular polygon based code construction has minimum distance that is higher at least by two in comparison with the dual regular polygon code of the same rate and length. A new minimum-distance bound is presented for codes from nonregular polygons of even diameter and equal number of points and lines. Finally, we prove that all codes derived from finite classical generalized quadrangles are quasi-cyclic and we give the explicit size of the circulant blocks in the parity-check matrix. Our simulation studies of several generalized polygon LDPC codes demonstrate powerful bit-error-rate (BER) performance when decoding is carried out via low-complexity variants of belief propagation.     

Systematic recursive construction of LDPC codes

This letter presents a systematic and recursive method to construct good low-density parity-check (LDPC) codes, especially those with high rate. The proposed method uses a parity check matrix of a quasi-cyclic LDPC code with given row and column weights as a core upon which the larger code is recursively constructed with extensive use of pseudorandom permutation matrices. This construction preserves the minimum distance and girth properties of the core matrix and can generate either regular, or irregular LDPC codes. The method provides a unique representation of the code in compact notation.     

Finite field construction for quasi-cyclic LDPC convolutional codes with cyclic 2-D MDS codes

The parity-check matrices for quasi-cyclic low-density parity-check convolutional (QC-LDPC-C) codes have different characteristics of time-varying periodicity and need to realize fast encoding. The finite field construction method for QC-LDPC-C codes with cyclic two-dimensional maximum distance separable (2-D MDS) codes is proposed using the base matrix framework and matrix unwrapping, thus the constructed parity-check matrices are free of length-4 cycles. The unwrapped matrices are constructed respectively based on different cyclic 2-D MDS codes for the case of matrix period less than or greater than constraint block length, and construction examples are given. LDPC-C codes with different periodicity characteristics are compared with QC-LDPC-C codes constructed with the proposed method. Experimental results show that QC-LDPC-C codes with the proposed method outperform the other codes and have lower encoding and decoding complexity.

     

Quasi-cyclic LDPC codes for fast encoding

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.     

On the minimum pseudo-codewords of LDPC codes

In this letter, we study the minimum pseudo-codewords of low-density parity-check (LDPC) codes under linear programming (LP) decoding. We show that a lower bound of Chaichanavong and Siegel on the pseudo-weight of a pseudo-codeword is tight if and only if this pseudo-codeword is a real multiple of a codeword. Using this result we further show that for some LDPC codes, e.g., Euclidean plane and projective plane LDPC codes, there are no other minimum pseudo-codewords except the real multiples of minimum codewords.     

Construction of LDPC codes over GF（q） with modified progressive edge growth

A parity check matrix construction method for constructing a low-density parity-check (LDPC) codes over GF(q) (q>2) based on the modified progressive edge growth (PEG) algorithm is introduced. First, the nonzero locations of the parity check matrix are selected using the PEG algorithm. Then the nonzero elements are defined by avoiding the definition of subcode. A proof is given to show the good minimum distance property of constructed GF(q)-LDPC codes. Simulations are also presented to illustrate the good error performance of the designed codes.     

Quasi-cyclic LDPC codes using overlapping matrices and their layered decoders

Quasi-cyclic (QC) low-density parity-check (LDPC) codes have the parity-check matrices consisting of circulant matrices. Since QC LDPC codes whose parity-check matrices consist of only circulant permutation matrices are difficult to support layered decoding and, at the same time, have a good degree distribution with respect to error correcting performance, adopting multi-weight circulant matrices to parity-check matrices is useful but it has not been much researched. In this paper, we propose a new code structure for QC LDPC codes with multi-weight circulant matrices by introducing overlapping matrices. This structure enables a system to operate on dual mode in an efficient manner, that is, a standard QC LDPC code is used when the channel is relatively good and an enhanced QC LDPC code adopting an overlapping matrix is used otherwise. We also propose a new dual mode parallel decoder which supports the layered decoding both for the standard QC LDPC codes and the enhanced QC LDPC codes. Simulation results show that QC LDPC codes with the proposed structure have considerably improved error correcting performance and decoding throughput.