Partial geometries form an interesting branch in combinatorial mathematics, and recently they have been shown to be very effective in the construction of LDPC codes with distinct geometric and algebraic structures. This paper presents three specific cyclic classes of partial geometries. Based on these three classes of partial geometries, three classes of LDPC codes and three classes of constant-weight codes are constructed. Codes in these two categories are either cyclic or quasi-cyclic. Designs and constructions of these codes are straightforward and flexible without a need for extensive computer search. It is shown that long high-rate LDPC codes constructed based on the three classes of cyclic partial geometries perform well over the additive white Gaussian noise channel (AWGNC) with iterative decoding algorithm based on belief propagation. They can achieve low error-rates without visible error-floor and their decoding converges rapidly. These LDPC codes also perform well over the binary erasure channel (BEC) and are very effective in correcting phased-bursts of erasures. Based on cyclic partial geometries, a special type of constant-weight codes, called balanced codes, can be constructed. The constant-weight codes constructed based on partial geometries are either optimal or nearly optimal.

In his pioneering work on LDPC codes, Gallager dismissed codes with parity-check matrices of weight two after proving that their minimum Hamming distances grow at most logarithmically with their code lengths. In spite of their poor minimum Hamming distances, it is shown that quasi-cyclic LDPC codes with parity-check matrices of column weight two have good capability to correct phased bursts of erasures which may not be surpassed by using quasi-cyclic LDPC codes with parity-check matrices of column weight three or more. By modifying the parity-check matrices of column weight two and globally coupling them, the erasure correcting capability can be further enhanced. Quasi-cyclic LDPC codes with parity-check matrices of column weight three or more that can correct phased bursts of erasures and perform well over the AWGN channel are also considered. Examples of such codes based on Reed-Solomon and Gabidulin codes are presented.

In this paper, we propose a new approach to design finite alphabet iterative decoders (FAIDs) for Low-Density Parity Check (LDPC) codes over binary symmetric channel (BSC) via recurrent quantized neural networks (RQNN). We focus on the linear FAID class and use RQNNs to optimize the message update look-up tables by jointly training their message levels and RQNN parameters. Existing neural networks for channel coding work well over Additive White Gaussian Noise Channel (AWGNC) but are inefficient over BSC due to the finite channel values of BSC fed into neural networks. We propose the bit error rate (BER) as the loss function to train the RQNNs over BSC. The low precision activations in the RQNN and quantization in the BER cause a critical issue that their gradients vanish almost everywhere, making it difficult to use classical backward propagation. We leverage straight-through estimators as surrogate gradients to tackle this issue and provide a joint training scheme. We show that the framework is flexible for various code lengths and column weights. Specifically, in high column weight case, it automatically designs low precision linear FAIDs with superior performance, lower complexity, and faster convergence than the floating-point belief propagation algorithms in waterfall region.

A novel scheme is presented for encoding and iterative soft-decision decoding of cyclic codes of prime lengths. The encoding of a cyclic code of a prime length is performed on a collection of codewords which are mapped through Galois Fourier transform into a codeword in a low-density parity-check code with a binary parity-check matrix for transmission. Using this matrix, binary iterative soft-decision decoding algorithm is applied to jointly decode a collection of codewords from the cyclic code. The joint-decoding allows for information sharing among the received vectors corresponding to the codewords in the collection during the iterative decoding process. For decoding Reed-Solomon and BCH codes of prime lengths, the proposed decoding scheme not only requires much lower decoding complexity than other soft-decision decoding algorithms for these codes, but also yields superior performance. The proposed decoding scheme can also achieve a joint-decoding gain over the maximum likelihood decoding of individual codewords. The decoding scheme is also applied to quadratic residue codes.

In this paper, we introduce a method of using quantized neural networks (QNN) to design finite alphabet message passing decoders (FAID) for Low-Density Parity Check (LDPC) codes. Specifically, we construct a neural network with low precision activations to optimize a FAID over Additive White Gaussian Noise Channel (AWGNC). The low precision activations cause a critical issue that their gradients vanish almost everywhere, making it difficult to use classical backward propagation. We introduce straight-through estimators (STE) to avoid this problem, by replacing zero derivatives of quantized activations with surrogate gradients in the chain rules. We present a systematic approach to train such networks while minimizing the bit error rate, which is a widely used and accurate metric to measure the performance of iterative decoders. Examples and simulations show that by training a QNN, a FAID with 3-bit of message and 4-bit of channel output can be obtained, which performs better than the more complex floating-point minsum decoding algorithm. This methodology is promising in the sense that it facilitates designing low-precision FAID for LDPC codes while maintaining good error performance in a flexible and efficient manner.

This paper presents a construction of a class of partial geometries based on RS codes of prime lengths and shows that LDPC codes constructed based on Reed-Solomon codes of prime lengths are finite geometry LDPC codes. Furthermore, a new method for design and construction of nonbinary quasi-cyclic LDPC codes based on the conventional parity-check matrices of Reed-Solomon codes is presented. Simulation results show that the constructed nonbinary LDPC codes perform well over the additive white Gaussian channel.

This paper presents designs and constructions of two classes of binary quasi-cyclic LDPC codes for correcting multiple random phased-bursts of erasures over the binary erasure channel. The erasure correction of codes in both classes is characterized by the cycle and adjacency structure of their Tanner graphs. Erasure correction of these codes is a very simple process which requires only modulo-2 additions. The codes in the second class are capable of correcting locally and globally distributed phased-bursts of erasures with a two-phase iterative erasure-correction process.

Designs and constructions of quasi-cyclic (QC) LDPC codes for the AWGN channel are presented. The codes are constructed based on the conventional parity-check matrices of Reed-Solomon (RS) codes and are referred to as RS-QC-LDPC codes. Several classes of RS-QC-LDPC codes are given. Cycle structural properties of the Tanner graphs of codes in these classes are analyzed and specific methods for constructing codes with girth at least eight and reducing their short cycles are presented. The designed codes perform well in both waterfall and low error-rate regions.

In this paper, two globally-coupled low-density parity check (GC-LDPC) code constructing methods are presented. The first is a Reed-Solomon-like GC-LDPC code that eases the design parameter searching in algebraic code constructions. The parameters such as the finite field, the associated prime factors, or the number of local codes can be more easily determined. The directly masking approach, the second proposed scheme, facilitates flexible structure designs of the parity check matrices. The two approaches can be regarded as generalizations of the original GC-LDPC codes. The design examples and the simulation results show the feasibility of the proposed techniques. In addition to code constructions, some applications and usage scenarios of GC-LDPC codes are addressed as well.

A number of recent works have used a variety of combinatorial constructions to derive Tanner graphs for LDPC codes and some of these LDPC codes have been shown to perform well in terms of their probability of errors and error floors. Such graphs are bipartite and many of these constructions yield biregular graphs, where the degree of left vertices is a constant c + 1 and that of the right vertices is a constant d + 1. Such graphs are termed (c+1, d +1) biregular bipartite graphs. Two properties of interest in such work is the girth of the graph and the number of short cycles in the graph, cycles of length either the girth or slightly larger. Such numbers have been shown to be related to the error floor of the probability of error curve of the related LDPC code. Using known results of graph theory, it is shown how the girth and the number of cycles of length equal to the girth may be computed for these (c + 1, d + 1) biregular bipartite graphs knowing only the parameters c and d and the numbers of left and right vertices. While numerous algorithms to determine the number of short cycles in arbitrary graphs exist, the reduction of the problem from an algorithm to a computation for these biregular bipartite graphs is of interest.

This paper presents a special type of nonbinary LDPC codes which are constructed based on Reed-Solomon codes. For a code of this type, its Tanner graph is composed of a set of disjoint and identical Tanner graphs, which are coupled together by a group of global check-nodes. Such a code is called a globally coupled LDPC code. This type of codes are capable of correcting random symbol errors, multiple phased bursts of erasures, and a single long burst of erasures.

A novel scheme is presented for encoding and decoding of Reed-Solomon codes of prime lengths. Encoding is performed on a collection of codewords which are mapped through Galois Fourier transform into a codeword in a low-density parity-check code with a binary parity-check matrix for transmission. Using this matrix, a binary iterative soft-decision decoding algorithm is applied to jointly decode a collection of codewords in the Reed-Solomon code. By allowing information sharing among the received vectors corresponding to the code-words in the collection, the proposed decoding scheme achieves superior performance over algorithms decoding individual Reed-Solomon codewords including maximum likelihood decoding.

The use of partial geometries to construct parity-check matrices for binary low-density parity-check (LDPC) codes has resulted in the design of successful codes with a probability of error on the AWGN channel close to the Shannon capacity at bit error rate down to $10^{-15}$ . Such considerations have motivated this further investigation. A new and simple construction of a type of partial geometries with a quasi-cyclic (QC) structure is given and their properties are investigated. Two new classes of this type of partial geometries, one based on prime fields and the other based on cyclic subgroups of prime orders of finite fields, are constructed. QC-LDPC codes with good error performances are constructed based on these two new classes of partial geometries. The trapping sets of the partial geometry codes were previously considered using the geometric aspects of the underlying structure to derive information on the size of allowable trapping sets. This topic is further considered here. Finally, there is a natural relationship between partial geometries and strongly regular graphs. The eigenvalues of the adjacency matrices of such graphs are well known, and it is of interest to determine if any of the Tanner graphs derived from the partial geometries are good expanders for certain parameter sets, since it can be argued that codes with good geometric and expansion properties might perform well on the AWGN channel under message-passing decoding.

Fire codes are cyclic codes generated by the product of two polynomials: a binomial that characterizes the code's guaranteed burst-correcting capability and an irreducible polynomial that characterizes the code length. However, the true burst-correcting capability of a Fire code may exceed its guaranteed burst-correcting capability, which can be thought of as the designed burst-correcting capability of the Fire code. The true burst-correcting capability of a Fire code depends on the irreducible polynomial used in code construction. In this paper, the maximum true burst-correcting capabilities of Fire codes are considered. Fire codes are classified based on three parameters: their designed burst-correcting capabilities, the least common multiple of the periods of the binomials, and the irreducible polynomials used in their constructions, and the ratios of the periods of primitive polynomials of the same degrees as the irreducible polynomials to the periods of the irreducible polynomials. It is shown that the maximum true burst-correcting capability of each class, which pertains to an infinite number of codes, can be determined by checking whether or not a finite number of incongruences have a solution. It is also shown that in each class, there is an infinite number of Fire codes, with increasing code lengths, that attain this maximum. The maximum true burst-correcting capabilities of several classes of Fire codes are determined. It is also shown that there are infinite sequences of irreducible polynomials that generate cyclic codes of rates approaching one and with burst-correcting capabilities that exceed any given number.

The Rosenbloom-Tsfasman metric has attracted the attention of many researchers as a generalization of the Hamming metric that is relevant to practical problems. Codes for this metric were considered. In particular, Reed-Solomon codes were generalized to be compatible with this metric. In this paper, a generalization of BCH codes for the Rosenbloom-Tsfasman metric is proposed. This generalization is based on considering BCH codes as subfield subcodes of Reed-Solomon codes. By characterizing these subfield subcodes, an explicit construction of BCH codes for the Rosenbloom-Tsfasman metric is provided. Two important properties of Reed-Solomon codes and BCH codes for the Rosenbloom-Tsfasman metric are studied and compared with those for the Hamming metric. These properties are cyclic structure and duality. The approach is based on Galois-Fourier transforms associated with Hasse derivatives.

Recently, Karimi and Banihashemi proposed an algorithm based on message-passing to count cycles in a graph of lengths less than double its girth. The algorithm uses only integer additions and subtractions to compute messages at the nodes of the graph that are passed to adjacent nodes. The complexity of the algorithm, when applied to a bipartite graph of girth g that has E edges, is O(gE2). The algorithm is superior to many other existing algorithms in the literature. In this paper, an improvement of this algorithm is presented that cuts both the complexity and the computing time by a factor of two. The improved algorithm is also applied to Tanner graphs of quasi-cyclic codes and, in this case, the complexity can be further cut by a factor of p, where p is the size of the circulants in the parity-check matrix of the quasi-cyclic code.

Polar codes proposed by Arikan are based on a linear kernel of dimension two with exponent 0.5. In this paper, binary kernels of maximum exponents of dimensions up to 16 are presented except for the case of dimension 12 where the maximum exponent is shown to be attained by either a constructed linear kernel or a possible nonlinear kernel with a specified partial distance sequence. The results show that the minimum dimension for which there exists a kernel with exponent greater than 0.5, i.e., exceeds the exponent of the linear kernel proposed by Arikan, is 14. For dimensions 14, 15, 16, discussed by Presman et al., along with 13, there are nonlinear kernels with exponents larger than any of that of a linear kernel. The kernels of these dimensions that have maximum exponent, although nonlinear over GF(2), are ℤ4-linear or ℤ2ℤ4-linear.

Fire codes are cyclic codes generated by the product of two polynomials: a binomial that characterizes the code's guaranteed burst correcting capability and an irreducible polynomial that characterizes the code length. However, the true burst correcting capability of a Fire code may exceed its guaranteed burst correcting capability. The true burst correcting capability of primitive Fire codes, in which the irreducible polynomial is primitive, is studied. In particular, the true burst correcting capability maximized over all primitive Fire codes with a given guaranteed burst correcting capability and a given greatest common divisor of the periods of the two factors of its generator polynomial is considered. It is shown that this maximum is attained by an infinite number of such codes.

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.

Polar codes are constructed based on kernels with polarizing properties. The performance of a polar code is characterized asymptotically in terms of the exponent of its kernel. The pioneering work of Arıkan on polar codes is based on a linear kernel of dimension two and exponent 0.5. In this paper, constructions of linear and nonlinear binary kernels of dimensions up to 16 are presented. The kernels are obtained using computer search or by shortening longer kernels obtained by computer search and a computer program is used to determine their exponents. It is proved that the constructed kernels have maximum exponents except in the case of nonlinear kernels of dimension 12 where it is demonstrated that the maximum exponent either equals that of the presented construction or assumes another specified value. The results show that the minimum dimension for which there exists a linear kernel with exponent greater than 0.5, i.e., exceeds the exponent of the linear kernel proposed by Arıkan, is 15, while this minimum dimension is 14 for nonlinear kernels. Furthermore, it is shown that there is a linear kernel with maximum exponent up to dimension 11. For dimensions 13, 14, 15, and 16, there are nonlinear kernels with exponents larger than any of that of a linear kernel. The kernels of these dimensions that have maximum exponent, although nonlinear over GF(2), are ℤ4-linear or ℤ2ℤ4-linear.

This paper presents a reduced-complexity iterative scheme and an algorithm for decoding non-binary quasi-cyclic (QC) LDPC codes of a specific type. The proposed decoding scheme and the algorithm together significantly reduce the hardware implementation complexity of a decoder with no performance degradation. Also presented in the paper is a simple method for constructing a class of non-binary QC-LDPC codes.

This paper presents a simple and flexible method for constructing QC-LDPC codes based on two arbitrary sets of a finite field. Based on this method, a high-rate, high-performance and very low error-floor QC-LDPC code is first constructed and then a class of rate-1/2 QC-LDPC codes whose Tanner graphs have girth 8 or larger is presented. Also presented is a reduced-complexity iterative decoding algorithm for QC-LDPC codes.

This paper presents a simple and very flexible method for constructing quasi-cyclic (QC) low density paritycheck (LDPC) codes based on finite fields. The code construction is based on two arbitrary subsets of elements from a given field. Some well known constructions of QC-LDPC codes based on finite fields and combinatorial designs are special cases of the proposed construction. The proposed construction in conjunction with a technique, known as masking, results in codes whose Tanner graphs have girth 8 or larger. Experimental results show that codes constructed using the proposed construction perform well and have low error-floors. Also presented in the paper is a reduced-complexity iterative decoding scheme for QC-LDPC codes based on the section-wise cyclic structure of their parity-check matrices. The proposed decoding scheme is an improvement of an earlier proposed reduced-complexity iterative decoding scheme.

It is shown that Euclidean geometry LDPC codes in conjunction with their shortened codes obtained by puncturing their parity-check matrices are effective in correcting combinations of errors and erasures with a two-phase decoding scheme. This is due to the large row redundancies of the parity-check matrices of these codes which are given by the incidence matrices of Euclidean geometries.

An effective reduced-complexity min-sum algorithm for decoding algebraic quasi-cyclic LDPC codes is presented. The proposed decoding algorithm significantly reduces the hardware implementation complexity, the size of memory required to store information, and the computational complexity of a decoder with no or a small loss in performance compared to the scaled min-sum algorithm.

Cyclic and quasi-cyclic algebraic LDPC codes constructed based on finite fields, finite geometries, and combinatorial designs can achieve excellent performance in terms of error rate, error floor and rate of decoding convergence with iterative decoding. However, the relatively high density of the parity-check matrix of an algebraic cyclic or quasi-cyclic LDPC code makes the hardware implementation complexity of the decoder quite large, which may be a critical issue in practical applications. This paper presents an effective reduced-complexity algorithm for decoding algebraic cyclic and quasi-cyclic LDPC codes based on the block cyclic structure and cyclic grouping of the rows of their parity-check matrices. The decoding of a code is carried out based on a single small submatrix of the parity-check matrix of the code in a revolving manner. The proposed decoding algorithm significantly reduces the hardware implementation complexity and the size of memory required to store information.

Many known constructions of LDPC codes can be placed in a general framework using the notion of partial geometries. Based on this notion, the structure of such LDPC codes can be analyzed using a geometric approach that illuminates important properties of their parity-check matrices. In this approach, trapping sets are represented by subgeometries of the geometry used to construct the code. Based on the incidence relations between lines and points in this geometry, the structure of trapping sets is investigated. On the other hand, it is shown that removing a subgeometry corresponding to a trapping set gives a punctured matrix which can be used as a parity-check matrix of an LDPC code. This relates trapping sets, represented by subgeometries, and punctured matrices, represented by the residual geometries. The null spaces of these punctured matrices are LDPC codes which inherit many of the good structural properties of the original code. Hence, new LDPC codes, with various lengths and rates, can be obtained by puncturing an LDPC code constructed based on a partial geometry. Furthermore, these punctured matrices and codes can be used in a two-phase decoding scheme to correct combinations of errors and erasures.

The trapping set structure of LDPC codes constructed using finite geometries is analyzed. A trapping set is modeled as a sub-geometry of the geometry used to construct an LDPC code. The variable nodes of a trapping set are viewed as points of the geometry and the check nodes adjacent to the variable nodes are viewed as the lines passing through any of these points. Based on this geometrical representation of a trapping set, its configuration can be determined.

A matrix-theoretic approach for studying quasi-cyclic codes based on matrix transformations via Fourier transforms and row and column permutations is developed. 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 amicable to the analysis and construction of quasi-cyclic low-density parity-check codes since it takes into account the specific parity-check matrix used for decoding with iterative message-passing algorithms. Based on this approach, the dimension of the codes and parity-check matrices for the dual codes can be determined. Several algebraic and geometric constructions of quasi-cyclic codes are presented as applications along with simulation results showing their performance over additive white Gaussian noise channels decoded with iterative message-passing algorithms.

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.

Non-binary low-density parity-check (NB-LDPC) codes can achieve better error-correcting performance than their binary counterparts at the cost of higher decoding complexity when the codeword length is moderate. The recently developed iterative reliability-based majority-logic NB-LDPC decoding has better performance-complexity tradeoffs than previous algorithms. This paper first proposes enhancement schemes to the iterative hard reliability-based majority-logic decoding (IHRB-MLGD). Compared to the IHRB algorithm, our enhanced (E-)IHRB algorithm can achieve significant coding gain with small hardware overhead. Then low-complexity partial-parallel NB-LDPC decoder architectures are developed based on these two algorithms. Many existing NB-LDPC code construction methods lead to quasi-cyclic or cyclic codes. Both types of codes are considered in our design. Moreover, novel schemes are developed to keep a small proportion of messages in order to reduce the memory requirement without causing noticeable performance loss. In addition, a shift-message structure is proposed by using memories concatenated with variable node units to enable efficient partial-parallel decoding for cyclic NB-LDPC codes. Compared to previous designs based on the Min-max decoding algorithm, our proposed decoders have at least tens of times lower complexity with moderate coding gain loss.

In this paper, a class of product codes based on low-density parity check (LDPC) constituent codes are constructed for multiple phased-burst erasure correction (MPBEC). These codes are shown to correct one large burst and/or a number of shorter bursts. A simple novel recursive erasure correction algorithm is proposed based on a recently discovered zero-span approach to linear block code analysis that can produce very powerful MPBEC capabilities. Analysis and data on how these codes work for additive white Gaussian noise (AWGN) channel are also presented.

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.

THIS PAPER IS ELIGIBLE FOR THE STUDENT PAPER AWARD. This paper analyzes trapping set structure of binary regular LDPC codes whose parity-check matrices satisfy the constraint that no two rows (or two columns) have more than one place where they both have non-zero components, which is called row-column (RC) constraint. For a (γ,ρ)-regular LDPC code whose parity-check matrix satisfies the RC-constraint, its Tanner graph contains no (κ, τ) trapping set with size κ ≤ γ and number τ of odd degree check nodes less than γ. For several classes of RC-constrained regular LDPC codes constructed algebraically, we show that their Tanner graphs contain no trapping sets of sizes smaller than their minimum weights.

Error-floors are the main reason for excluding LDPC codes from applications requiring very low bit-error rate. They are attributed to a particular structure in the codes' Tanner graphs, known as trapping sets, which traps the message-passing algorithms commonly used to decode LDPC codes, and prevents decoding from converging to the correct codeword. A technique is proposed to break trapping sets while decoding. Based on decoding results leading to a decoding failure, some bits are identified in a previous iteration and flipped and decoding is restarted. This backtracking may enable the decoder to get out of the trapped state. A semi-analytical method is also proposed to predict the error-floor after backtracking. Simulation results indicate the effectiveness of the proposed technique in lowering the error-floor. The technique, which has moderate complexity overhead, is applicable to any code without requiring a prior knowledge of the structure of its trapping sets.

Codes constructed based on finite geometries form a large class of cyclic codes with large minimum distances which can be decoded with simple majority-logic decoding in one or multiple steps. In 2001, Kou, Lin and Fossorier showed that the one-step majority-logic decodable finite geometry codes form a class of cyclic LDPC codes whose Tanner graphs are free of cycles of length 4. These cyclic finite geometry LDPC codes perform very well over the AWGN channel using iterative decoding based on belief propagation (IDBP) and have very low error-floors. However, the standard IDBP is not effective for decoding other cyclic finite geometry codes because their Tanner graphs contain too many short cycles of length 4 which severely degrade the decoding performance. This paper investigates iterative decoding of two-step majority-logic decodable finite geometry codes. Three effective algorithms for decoding these codes are proposed. These algorithms are devised based on the orthogonal structure of the parity-check matrices of the codes to avoid or reduce the degrading effect of the short cycles of length 4. These decoding algorithms provide significant coding gains over the standard IDBP using either the sum-product or the min-sum algorithms.

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.

Flash memory is a nonvolatile computer storage device which consists of blocks of cells. While increasing the voltage level of a single cell is fast and simple, reducing the level of a cell requires the erasing of the entire block containing the cell. Since block-erasures are costly, flash coding schemes have been developed to maximize the number of writes before a block-erasure is needed. A novel coding scheme based on error-correcting codes is presented that allows the cell levels to increase as evenly as possible and as a result, increases the number of writes before a block-erasure. The scheme is based on the premise that cells whose levels are higher than others need not be increased. This introduces errors in the recorded data which can be corrected by an error-correcting code provided that the number of erroneous cells is within the error-correcting capability of the code. The scheme is also capable of combating noise, causing additional errors and erasures, in flash memories in order to enhance data reliability. For added flexibility, the scheme can be combined with other flash codes to yield concatenated schemes of high memory rates.

Flash memory is a non-volatile computer storage device which consists of blocks of cells. While increasing the voltage level of a single cell is fast and simple, reducing the level of a cell requires the erasing of the entire block containing the cell. Since block erasures are costly, traditional flash coding schemes have been developed to maximize the number of writes before a block erasure is needed. A novel coding scheme based on error-correcting codes allows the cell levels to increase as evenly as possibly and as a result, increases the number of writes before a block erasure. The scheme is also capable of combating noise in flash memories in order to enhance data reliability.

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) and they outperform the corresponding pseudo-random QC-LDPC codes constructed with the PEG-algorithm. 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.

This paper presents a novel low-complexity iterative reliability-based decoding algorithm for LDPC codes over q-ary finite fields. This proposed algorithm has low complexity and hence provides an effective trade-off between error performance and decoding complexity compared to q-ary sum product algorithm. This decoding algorithm is devised based on simple orthogonal concept of one-step majority-logic decoding for q-ary linear block codes. It requires only integer and finite field operations and converges very fast in decoding. It is particularly effective for decoding LDPC codes constructed based on finite geometries and finite fields.

This paper shows that a cyclic code can be put into quasi-cyclic form by decomposing a circular parity-check matrix through column and row permutations. Such a decomposition of a circular parity-check matrix of a cyclic code produces a group of shorter cyclic or quasi-cyclic codes and leads to a new method for constructing long cyclic codes from short cyclic codes. Also in this paper, new classes of cyclic and quasi-cyclic LDPC codes are derived from cyclic Euclidean geometry LDPC codes by decomposing their circular parity-check matrices. These new LDPC codes perform well and enlarge the repertoire of cyclic and quasi-cyclic LDPC codes.

This paper presents a class of high-rate codes called low-density parity-check accumulate (LDPCA) codes. The code design is the serial concatenation of an LDPC outer code and an accumulator with an interleaver. The iterative decoding for the LDPCA code design has complexity linear to the code length. When using regular LDPC codes with column weight 2, the proposed codes have low encoding complexity and are advantageous for hardware implementation. Simulation results show that the regular LDPCA codes have the same error performance with the regular LDPC codes at the waterfall region and outperform product accumulate codes at the error floor region. The investigation on weight distributions proves that regular LDPCA codes have the asymptotic minimum distance proportional to the code length. In addition, iterative decoding thresholds under density evolution are obtained with a Gaussian approximation.

This paper presents two low-complexity reliability-based message-passing algorithms for decoding LDPC codes over non-binary finite fields. These two decoding algorithms require only finite field and integer operations and they provide effective trade-off between error performance and decoding complexity compared to the non-binary sum product algorithm. They are particularly effective for decoding LDPC codes constructed based on finite geometries and finite fields.

This paper presents two new large classes of QC-LDPC codes, one binary and one non-binary. Codes in these two classes are constructed by array dispersions of row-distance constrained matrices formed based on additive subgroups of finite fields. Experimental results show that codes constructed perform very well over the AWGN channel with iterative decoding based on belief propagation. Codes of a subclass of the class of binary codes have large minimum distances comparable to finite geometry LDPC codes and they offer effective tradeoff between error performance and decoding complexity when decoded with low-complexity reliability-based iterative decoding algorithms such as binary message passing decoding algorithms. Non-binary codes decoded with a Fast-Fourier Transform based sum-product algorithm achieve significantly large coding gains over Reed-Solomon codes of the same lengths and rates decoded with either the hard-decision Berlekamp-Massey algorithm or the algebraic soft-decision Kotter-Vardy algorithm. They have potential to replace Reed-Solomon codes in some communication or storage systems where combinations of random and bursts of errors (or erasures) occur.

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.

An error-burst correcting algorithm is developed based on a circulant parity-check matrix of a cyclic code. The proposed algorithm is more efficient than error trapping if the code rate is less than about 2/3. It is shown that for any (n, k) cyclic code, there is an n × n circulant parity-check matrix such that the algorithm, applied to this matrix, corrects error bursts of lengths up to the error-burst correction limit of the cyclic code. This same matrix can be used to efficiently correct erasure bursts of lengths up to n - k. The error-burst correction capabilities of a class of cyclic low-density parity-check (LDPC) codes constructed from finite geometries are also considered.

FPGAs are widely used for evaluating the error-floor performance of LDPC (low-density parity check) codes. We propose a scalable vector decoder for FPGA-based implementation of quasi-cyclic (QC) LDPC codes that takes advantage of the high bandwidth of the embedded memory blocks (called Block RAMs in a Xilinx FPGA) by packing multiple messages into the same word. We describe a vectorized overlapped message passing algorithm that results in 3.5times to 5.5times speedup over state-of-the-art FPGA implementations in literature.

In this paper, we investigate the asymptotic ensemble weight enumerators of multi-edge type codes whose component codes are arbitrary block codes. Two forms of asymptotic growth rate of codewords, corresponding to the primal and dual problems, are obtained. Furthermore, for the codewords of small linear-sized weights, we develop a simplification method to restrict the search space of the primal problem and study the optimality conditions of the dual problem, giving a first-order approximation of the growth rate and a condition of exponentially few small weight codewords.

This paper presents two novel reliability-based iterative majority-logic decoding algorithms for LDPC codes. Both algorithms are binary message-passing algorithms and require only logical operations and integer additions. Consequently, they can be implemented with simple combinational logic circuits. They either outperform or perform just as well as the existing weighted bit-flipping or other reliability-based iterative decoding algorithms for LDPC codes in error performance with a faster rate of decoding convergence and less decoding complexity. Compared to the sum-product algorithm for LDPC codes, they offer effective trade-offs between performance and decoding complexity.

A unified approach for constructing binary and nonbinary quasi-cyclic LDPC codes under a single framework is presented. Six classes of binary and nonbinary quasi-cyclic LDPC codes are constructed based on primitive elements, additive subgroups, and cyclic subgroups of finite fields. Numerical results show that the codes constructed perform well over the AWGN channel with iterative decoding.

This paper presents two algebraic methods for constructing high performance and efficiently encodable nonbinary quasi-cyclic LDPC codes based on arrays of special circulant permutation matrices and multi-fold array dispersions. Codes constructed based on these methods perform well over the AWGN and other types of channels with iterative decoding based on belief-propagation. Experimental results show that over the AWGN channel, these non-binary quasi-cyclic LDPC codes significantly outperform Reed-Solomon codes of the same lengths and rates decoded with either algebraic hard-decision Berlekamp-Massey algorithm or algebraic soft-decision Kötter- Vardy algorithm. Also presented in this paper is a class of asymptotically optimal LDPC codes for correcting bursts of erasures. Codes constructed also perform well over flat fading channels. Non-binary quasi-cyclic LDPC codes have a great potential to replace Reed-Solomon codes in some applications in communication environments and storage systems for combating mixed types of noises and interferences.

This paper presents algebraic methods for constructing high performance and efficiently encodable non-binary quasi-cyclic LDPC codes based on flats of finite Euclidean geometries and array masking. Codes constructed based on these methods perform very well over the AWGN channel. With iterative decoding using a fast Fourier transform based sum-product algorithm, they achieve significantly large coding gains over Reed-Solomon codes of the same lengths and rates decoded with either algebraic hard-decision Berlekamp-Massey algorithm or algebraic soft-decision Kotter-Vardy algorithm. Due to their quasi-cyclic structure, these non-binary LDPC codes on Euclidean geometries can be encoded using simple shift-registers with linear complexity. Structured non-binary LDPC codes have a great potential to replace Reed-Solomon codes for some applications in either communication or storage systems for combating mixed types of noise and interferences.

In iterative decoding of LDPC codes, trapping sets often lead to high error floors. In this work, we propose a two-stage iterative decoding to break trapping sets. Simulation results show that the error floor performance can be significantly improved with this decoding scheme.

This paper investigates error control at the physical layer of a multicast network using low-density parity-check (LDPC) codes. Packets for transmission are encoded into LDPC codewords. A joint iterative message-passing scheme for decoding LDPC codewords at a receive node in the network is proposed to improve error performance. Also proposed is a split-codeword transmission to provide equal error protection for all transmitted packets. Density evolution analysis and some simulation results are also presented.

This paper presents two new algebraic constructions of high performance non-binary quasi-cyclic LDPC codes based on array dispersions of matrices over non-binary fields. Codes constructed perform well over the AWGN channel with iterative decoding using aFastFourierTransformbased sum-product algorithm. They achieve significantly large coding gains over Reed-Solomon codes of the same lengths and rates decoded with either the hard-decision Berlekamp-Massey algorithm or the algebraic soft-decision Koetter-Vardy algorithm. Due to their quasi-cyclic structure, they can be efficiently encoded using simple shift-registers with linear complexity. They have a potential to replace RS codes for some applications in communication and storage systems.

This paper presents algebraic methods for constructing high performance quasi-cyclic LDPC codes over non-binary 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 sum-product 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 Kotter-Vardy algorithm, and the Jiang-Narayananpsilas 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.

This paper presents new methods for constructing efficiently encodable quasi-cyclic LDPC codes based on special balanced incomplete block designs (BIBD's). Codes constructed perform well over both the AWGN and binary erasure channels with iterative decoding.

This paper is concerned with construction of efficiently encodable nonbinary quasi-cyclic LDPC codes based on finite fields. Four classes of nonbinary quasi-cyclic LDPC codes are constructed. Experimental results show that codes constructed perform well with iterative decoding using a fast Fourier transform based q-ary sum-product algorithm and they achieve significant coding gains over Reed-Solomon codes of the same lengths and rates decoded with either algebraic hard- decision Berlekamp-Massey algorithm or algebraic soft-decision Kotter-Vardy algorithm.

This paper presents five methods for constructing nonbinary LDPC codes based on finite geometries. These methods result in five classes of nonbinary LDPC codes, one class of cyclic LDPC codes, three classes of quasi-cyclic LDPC codes and one class of structured regular LDPC codes. Experimental results show that constructed codes in these classes decoded with iterative decoding based on belief propagation perform very well over the AWGN channel and they achieve significant coding gains over Reed-Solomon codes of the same lengths and rates with either algebraic hard-decision decoding or Kotter-Vardy algebraic soft-decision decoding at the expense of a larger decoding computational complexity.

This paper presents new methods for efficiently constructing encodable quasi-cyclic low-density parity-check (LDPC) codes based on special balanced incomplete block designs (BIBDs). Codes constructed perform well over both the additive white Gaussian noise (AWGN) and binary erasure channels with iterative decoding.

In the late 1950s and early 1960s, finite fields were successfully used to construct linear block codes, especially cyclic codes, with large minimum distances for hard-decision algebraic decoding, such as Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes. This paper shows that finite fields can also be successfully used to construct algebraic low-density parity-check (LDPC) codes for iterative soft-decision decoding. Methods of construction are presented. LDPC codes constructed by these methods are quasi-cyclic (QC) and they perform very well over the additive white Gaussian noise (AWGN), binary random, and burst erasure channels with iterative decoding in terms of bit-error probability, block-error probability, error-floor, and rate of decoding convergence, collectively. Particularly, they have low error floors. Since the codes are QC, they can be encoded using simple shift registers with linear complexity.

Two algebraic methods for systematic construction of structured regular and irregular low-density parity-check (LDPC) codes with girth of at least six and good minimum distances are presented. These two methods are based on geometry decomposition and a masking technique. Numerical results show that the codes constructed by these methods perform close to the Shannon limit and as well as random-like LDPC codes. Furthermore, they have low error floors and their iterative decoding converges very fast. The masking technique greatly simplifies the random-like construction of irregular LDPC codes designed on the basis of the degree distributions of their code graphs

This paper investigates cyclic codes for correcting bursts of errors from a new point of view. A simple iterative algorithm for correcting bursts of errors is developed. This algorithm is optimal in the sense that it corrects burst of errors of lengths up to the burst-error-correction limit of a cyclic code. Also included in the paper is an iterative process for correcting bursts of erasures.

This paper presents a general and three specific algebraic methods for constructing efficiently encodable non-binary 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.

LDPC codes have excellent performance when iteratively decoded based on sparse parity-check matrices over both the AWGN channel and the erasure channel. In this paper, we propose a simple burst-correction decoding scheme for cyclic LDPC codes based on the same sparse circulant parity-check matrices used to perform iterative decoding. Although the proposed scheme may not achieve the full burst-correcting capability of the codes, it is rather simple and fast. We also show that the full burst-correcting capabilities of some codes constructed from Euclidean and projective geometries are very close to the Reiger upper bound

This paper presents three algebraic methods for constructing nonbinary quasi-cyclic (QC)-LDPC codes. Three classes of efficiently encodable QC-LDPC codes over nonbinary finite fields are constructed. Experimental results show that constructed codes decoded with iterative decoding perform well over the AWGN channel and they achieve significant coding gains over Reed-Solomon (RS) codes of the same lengths and rates decoded with algebraic decoding

Quasi-cyclic (QC) low-density parity-check (LDPC) codes form an important subclass of LDPC codes. These codes have encoding advantage over other types of LDPC codes. This paper addresses the issue of efficient encoding of QC-LDPC codes. Two methods are presented to find the generator matrices of QC-LDPC codes in systematic-circulant (SC) form from their parity-check matrices, given in circulant form. Based on the SC form of the generator matrix of a QC-LDPC code, various types of encoding circuits using simple shift registers are devised. It is shown that the encoding complexity of a QC-LDPC code is linearly proportional to the number of parity bits of the code for serial encoding, and to the length of the code for high-speed parallel encoding.

This paper is concerned with construction of quasi-cyclic (QC) low-density parity-check (LDPC) codes for three different types of channels: the additive white Gaussian noise, the binary random erasure, and the binary burst erasure channels. Two algebraic methods for systematic construction of QC-LDPC codes are presented. Codes constructed perform well over all three types of channels

This paper presents methods for efficient encoding of quasi-cyclic LDPC codes. Based on these methods, encoding of quasi-cyclic LDPC codes can be implemented using simple shift-registers with complexity linearly proportional to the number of parity-check bits of a code for serial encoding and to the length of a code for parallel encoding. Various encoding circuits are devised and they provide a range of trade-offs between encoding complexity and speed.

This paper presents three algebraic methods for constructing q-ary LDPC codes. The first method gives a class of dispersed Reed-Solomon codes as LDPC codes. The second method gives a class of q-ary quasi-cyclic LDPC codes. The third method gives two classes of q-ary finite geometry LDPC codes. Codes constructed by these methods perform very well with iterative decoding, even for short codes.

This paper presents two algebraic methods for constructing efficiently encodable quasi-cyclic (QC) LDPC codes that perform well on both the AWGN and binary erasure channels with iterative decoding in terms of bit-error performance, block error performance and error-floor, collectively. The constructions are based on the cyclic subgroups of the multiplicative groups of finite fields and affine mappings

This paper presents a method for constructing structured LDPC codes based on finite fields and affine mappings. The Tanner graphs of these codes have girth at least six. Experimental results show that codes constructed based on this method with iterative decoding perform well on both the AWGN and binary erasure channels. Furthermore, codes constructed based on prime fields are quasi-cyclic.

Efficient Encoding of Quasi-Cyclic Low-Density Parity-Check Codes Quasi-cyclic (QC) low-density parity-check (LDPC) codes form an important subclass of LDPC codes. These codes have encoding advantage over other types of LDPC codes. This paper addresses the issue of efficient encoding of QC-LDPC codes. Two methods are presented to find the generator matrices of QC-LDPC codes in systematic-circulant form from their parity-check matrices given in circulant form. Based on the systematic-circulation form of the generator matrix of a QC-LDPC code, various types of encoding circuits using simple shift registers are devised. It is shown that the encoding complexity of a QC-LDPC code is linearly proportional to the number of parity bits of the code for serial encoding, and to the length of the code for high-speed parallel encoding.

This paper presents a superposition method for constructing low-density parity-check (LDPC) codes. Several classes of structured LDPC codes are constructed. Codes in these classes perform well with iterative decoding, and their Tanner graphs have girth at least six.

New algebraic methods for constructing codes based on hyperplanes of two different dimensions in finite geometries are presented. The new construction methods result in a class of multistep majority-logic decodable codes and three classes of low-density parity-check (LDPC) codes. Decoding methods for the class of majority-logic decodable codes, and a class of codes that perform well with iterative decoding in spite of having many cycles of length 4 in their Tanner graphs, are presented. Most of the codes constructed can be either put in cyclic or quasi-cyclic form and hence their encoding can be implemented with linear shift registers.

This letter presents two classes of quasi-cyclic low-density parity-check codes that perform close to the Shannon limit.

This paper presents a two-stage turbo-coding scheme for Reed-Solomon (RS) codes through binary decomposition and self-concatenation. In this scheme, the binary image of an RS code over GF(2/sup m/) is first decomposed into a set of binary component codes with relatively small trellis complexities. Then the RS code is formatted as a self-concatenated code with itself as the outer code and the binary component codes as the inner codes in a turbo-coding arrangement. In decoding, the inner codes are decoded with turbo decoding and the outer code is decoded with either an algebraic decoding algorithm or a reliability-based decoding algorithm. The outer and inner decoders interact during each decoding iteration. For RS codes of lengths up to 255, the proposed two-stage coding scheme is practically implementable and provides a significant coding gain over conventional algebraic and reliability-based decoding algorithms.

This correspondence presents a method for constructing structured regular low-density parity-check (LDPC) codes based on a special type of combinatoric designs, known as balance incomplete block designs. Codes constructed by this method have girths at least 6 and they perform well with iterative decoding. Furthermore, several classes of these codes are quasi-cyclic and hence their encoding can be implemented with simple feedback shift registers.

This correspondence presents three algebraic methods for constructing low-density parity-check (LDPC) codes. These methods are based on the structural properties of finite geometries. The first method gives a class of Gallager codes and a class of complementary Gallager codes. The second method results in two classes of circulant-LDPC codes, one in cyclic form and the other in quasi-cyclic form. The third method is a two-step hybrid method. Codes in these classes have a wide range of rates and minimum distances, and they perform well with iterative decoding.

The paper presents two classes of quasi-cyclic low-density parity-check (LDPC) codes which perform close to the Shannon limit. The construction of these codes is based on decomposition of circulant matrices constructed from finite geometries.

The notion of product block codes, whose generator matrix is the tensor product of the constituent generator matrices, is well established. Typically, they have good performance and a decoding algorithm with complexity on the order of the complexity of the decoding algorithms of the constituent codes. There has been much recent attention on the construction of bipartite graphs for low density parity check codes whose parity check matrices are the incidence matrices of right versus left vertices of the graph. The relation of the properties of the incidence matrix to code performance is difficult to establish precisely, although some guidelines are available. Two types of incidence matrix constructions are given here that show promise. In the first instance, a combinatorial construction is given, and, secondly, two types of graph products are considered for their application to LDPC codes.

The paper presents a computationally efficient hybrid reliability-based decoding algorithm for Reed-Solomon (RS) codes. This hybrid decoding algorithm consists of two major components, a re-encoding process and a successive erasure-and-error decoding process for both bit and symbol levels. The re-encoding process is to generate a sequence of candidate codewords based on the information provided by the codeword decoded by an algebraic decoder and a set of test error patterns. Two criteria are used for testing in the decoding process to reduce the decoding computational complexity. The first criterion is devised to reduce the number of re-encoding operations by eliminating the unlikely error patterns. The second criterion is to test the optimality of a generated candidate codeword. Numerical results show that the proposed decoding algorithm can achieve either a near-optimum error performance or an asymptotically optimum error performance.

The paper presents two efficient Viterbi decoding-based suboptimal algorithms for tailbiting codes. The first algorithm, the wrap-around Viterbi algorithm (WAVA), falls into the circular decoding category. It processes the tailbiting trellis iteratively, explores the initial state of the transmitted sequence through continuous Viterbi decoding, and improves the decoding decision with iterations. A sufficient condition for the decision to be optimal is derived. For long tailbiting codes, the WAVA gives essentially optimal performance with about one round of Viterbi trial. For short- and medium-length tailbiting codes, simulations show that the WAVA achieves closer-to-optimum performance with fewer decoding stages compared with the other suboptimal circular decoding algorithms. The second algorithm, the bidirectional Viterbi algorithm (BVA), employs two wrap-around Viterbi decoders to process the tailbiting trellis from both ends in opposite directions. The surviving paths from the two decoders are combined to form composite paths once the decoders meet in the middle of the trellis. The composite paths at each stage thereafter serve as candidates for decision update. The bidirectional process improves the error performance and shortens the decoding latency of unidirectional decoding with additional storage and computation requirements. Simulation results show that both proposed algorithms effectively achieve practically optimum performance for tailbiting codes of any length.

The statistical approach proposed by Agrawal and Vardy (see ibid., vol.46, no.1, p.60-83, 2000) to evaluate the error performance of the generalized minimum distance (GMD) decoding is extended to other reliability-based decoding algorithms for binary linear block codes, namely Chase (1972) type, combined GMD and Chase type, and order statistic decoding (OSD). In all cases, tighter and simpler bounds than those previously proposed have been obtained with this approach.

This paper presents a geometric approach to the construction of low-density parity-check (LDPC) codes. Four classes of LDPC codes are constructed based on the lines and points of Euclidean and projective geometries over finite fields. Codes of these four classes have good minimum distances and their Tanner (1981) graphs have girth 6. Finite-geometry LDPC codes can be decoded in various ways, ranging from low to high decoding complexity and from reasonably good to very good performance. They perform very well with iterative decoding. Furthermore, they can be put in either cyclic or quasi-cyclic form. Consequently, their encoding can be achieved in linear time and implemented with simple feedback shift registers. This advantage is not shared by other LDPC codes in general and is important in practice. Finite-geometry LDPC codes can be extended and shortened in various ways to obtain other good LDPC codes. Several techniques of extension and shortening are presented. Long extended finite-geometry LDPC codes have been constructed and they achieve a performance only a few tenths of a decibel away from the Shannon theoretical limit with iterative decoding.

Low density parity check (LDPC) codes with iterative decoding based on belief propagation (IDBP) achieve astonishing error performance close to the Shannon limit. Until now there has been no known method for constructing these Shannon limit approaching codes systematically. Good LDPC codes are largely generated by computer search. As a result, the encoding of long LDPC codes is in general very complex. This paper presents the first algebraic method for constructing LDPC codes systematically based on finite analytic geometries. Four classes of finite geometry LDPC codes with relatively good minimum distances are constructed. These codes are either cyclic or quasi-cyclic and therefore their encoding can be implemented with simple linear feedback shift registers. Long finite geometry LDPC codes have been constructed and they achieve an error performance only a few tenths of a dB away from the Shannon limit. Finite geometry LDPC codes are strong competitors to turbo codes for error control in communication and digital data storage systems.

In this letter, Chase decoding algorithms are generalized into a family of bounded distance decoding algorithms, so that the conventional Chase algorithm-2 and Chase algorithm-3 become the two extremes of this family. Consequently, more flexibility in the tradeoffs between error performance and decoding complexity is provided by this generalization, especially for codes with large minimum distance. Finally this approach is extended to decoding with erasures.

In this paper, multilevel coded asymmetric modulation with multistage decoding and unequal error protection (UEP) is discussed. These results further emphasize the fact that unconventional signal set partitionings are more promising than traditional (Ungerboeck-type) partitionings, to achieve UEP capabilities with multilevel coding and multistage decoding. Three types of unconventional partitionings are analyzed for asymmetric 8-PSK and 16-QAM constellations over the additive white Gaussian noise channel to introduce design guidelines. Generalizations to other PSK and QAM type constellations follow the same lines. Upper bounds on the bit-error probability based on union bound arguments are first derived. In some cases, these bounds become loose due to the large overlappings of decision regions associated with asymmetric constellations and unconventional partitionings. To overcome this problem, simpler and tighter approximated bounds are derived. Based on these bounds, it is shown that additional refinements can be achieved in the construction of multilevel UEP codes, by introducing asymmetries in PSK and QAM signal constellations.

The maximum a posterioriprobability (MAP) algorithm is a trellis-based MAP decoding algorithm. It is the heart of turbo (or iterative) decoding that achieves an error performance near the Shannon limit. Unfortunately, the implementation of this algorithm requires large computation and storage. Furthermore, its forward and backward recursions result in a long decoding delay. For practical applications, this decoding algorithm must be simplifled and its decoding complexity and delay must be reduced. In this paper, the MAP algorithm and its variation's, such as log-MAP and max-log-MAP algorithms, are first applied to sectionalized trellises for linear block codes and carried out as two-stage decodings. Using the structural properties of properly sectionalized trellises, the decoding complexity and delay of the MAP algorithms can be reduced. Computation-wise optimum sectionalizations of a trellis for MAP algorithms are investigated. Also presented in this paper are bidirectional and parallel MAP decodings.

Previously, the belief propagation (BP) algorithm has received a lot of attention in the coding community, mostly due to its near-optimum decoding for low-density parity check (LDPC) codes and its connection to turbo decoding. In this paper, we investigate the performance achieved by the BP algorithm for decoding one-step majority logic decodable (OSMLD) codes. The BP algorithm is expressed in terms of likelihood ratios rather than probabilities, as conventionally presented. The proposed algorithm fits better the decoding of OSMLD codes with respect to its numerical stability due to the fact that the weights of their check sums are often much higher than that of the corresponding LDPC codes. Although it has been believed that OSMLD codes are far inferior to LDPC codes, we show that for medium code lengths (say between 200-1000 bits), the BP decoding of OSMLD codes can significantly outperform BP decoding of their equivalent LDPC codes. The reasons for this behavior are elaborated.

In this paper, theoretical upper bounds and computer simulation results on the error performance of multilevel block coded modulations for unequal error protection (UEP) and multistage decoding are presented. It is shown that nonstandard signal set partitionings and multistage decoding provide excellent UEP capabilities beyond those achievable with conventional coded modulation. The coding scheme is designed in such a way that the most important information bits have a lower error rate than other information bits. The large effective error coefficients, normally associated with standard mapping by set partitioning, are reduced by considering nonstandard partitionings of the underlying signal set. The bits-to-signal mappings induced by these partitionings allow the use of soft-decision decoding of binary block codes. Moreover, parallel operation of some of the staged decoders is possible, to achieve high data rate transmission, so that there is no error propagation between these decoders. Hybrid partitionings are also considered that trade off increased intraset distances in the last partition levels with larger effective error coefficients in the middle partition levels. The error performance of specific examples of multilevel codes over 8-PSK and 64-QAM signal sets are simulated and compared with theoretical upper bounds on the error performance.

This paper investigates the principle of metric differences for trellis decoding of convolutional codes. Based on this differential method, a new algorithm, referred to as differential trellis decoding (DTD), is proposed. DTD offers an alternative to the conventional "add-compare-select" (ACS) method for implementing the Viterbi algorithm.

In this study, error performance analysis and simulation results are provided for the weighted erasure decoding (WED) of binary linear block codes on Q-ary output channel. In particular, the optimum channel quantizer for WED is derived and compared with WED designed based on the cutoff rate criterion, as traditionally proposed. Simulations are conducted for the (64, 42, 8) Reed-Muller (RM) code on 4-ary and 8-ary output channels. For Q=8 and the bit error rate (BER) 10/sup -4/, the WED with optimum quantizer has a 0.65 dB coding gain over the WED with cutoff rate quantizer and a 1.0 dB coding gain over algebraic decoding.

In general, a channel quantizer for a communication system subject to additive white Gaussian noise (AWGN) is designed based on the cutoff rate. This criterion is good if the scheme considered performs close to the theoretical performance corresponding to the cutoff rate, as for error control systems employing convolutional codes. However, it is no longer true for systems using low complexity suboptimum decoding algorithms for block codes. We illustrate this point and present three examples for which we compare the optimum quantizer and the quantizer based on the cutoff rate for Q=4 quantization levels.

This paper presents two simple and effective criteria for stopping the iteration process in turbo decoding with a negligible degradation of the error performance. Both criteria are devised based on the cross-entropy (CE) concept. They are as efficient as the CE criterion, but require much less and simpler computations.

In this correspondence, the low-weight terms of the weight distribution of the block code obtained by terminating a convolutional code after x information blocks are expressed as a function of x. It is shown that this function is linear in x for codes with noncatastrophic encoders, but quadratic in x for codes with catastrophic encoders. These results are useful to explain the poor performance of convolutional codes with a catastrophic encoder at low-to-medium signal-to-noise ratios.

This paper presents a new low-weight trellis-based soft-decision iterative decoding algorithm for binary linear block codes. The algorithm is devised based on a set of optimality conditions and the generation of a sequence of candidate codewords for an optimality test. The initial candidate codeword is generated by a simple decoding method. The subsequent candidate codewords, if needed, are generated by a chain of low-weight trellis searches, one at a time. Each search is conducted through a low-weight trellis diagram centered around the latest candidate codeword and results in an improvement over the previous candidate codewords that have been already tested. This improvement is then used as the next candidate codeword for a test of optimality. The decoding iteration stops whenever a candidate codeword is found to satisfy a sufficient condition on optimality or the latest low-weight trellis search results in a repetition of a previously generated candidate codeword. A divide-and-conquer technique is also presented for codes that are not spanned by their minimum-weight codewords. The proposed decoding algorithm has been applied to some well-known codes of lengths 48, 64, and 128. Simulation results show that the proposed algorithm achieves either practically optimal error performance for the example codes of length 48 and 64 or near optimal error performance for the (128, 29, 32) RM code with a significant reduction in computational decoding complexity.

In this article, constructions of generalized concatenated (GC) codes with good rates and distances are presented. Some of the proposed GC codes have simpler trellis complexity than Euclidean geometry (EG), Reed-Muller (RM), or Bose-Chaudhuri-Hocquenghem (BCH) codes of approximately the same rates and minimum distances, and in addition can be decoded with trellis-based multistage decoding up to their minimum distances. Several codes of the same length, dimension, and minimum distance as the best linear codes known are constructed.

This paper considers block-coded 8-phase-shift-keying (PSK) modulations for the unequal error protection (UEP) of information transmitted over Rayleigh-fading channels. Both conventional linear block codes and linear UEP (LUEP) codes are combined with a naturally labeled 8-PSK signal set, using the multilevel construction of Imai and Hirakawa (1977). Computer simulation results are presented showing that, over Rayleigh-fading channels, it is possible to improve the coding gain for the most significant bits with the use of binary LUEP codes as constituent codes, in comparison with using conventional binary linear codes alone.

This paper presents a unified method for evaluating the error-correction radii of many reliability-based soft-decision decoding algorithms for binary linear block codes. Based on this unified method, these decoding algorithms as well as their potential improvements can be compared directly. The error-correction radius for each of these decoding algorithms is determined by finding the closest point on the boundary of the decision region of the soft-decision decoder to the transmitted signal sequence. It is shown that this problem can be formulated as a constrained optimization problem. Although no general closed-form expression is possible, a simple algorithm that always converges to the optimum solution of this optimization problem is presented. Based on these results, the error-correction radii of some well-known reliability-based soft-decision decoding algorithms are revisited and compared.

In this correspondence, the bit-error probability P/sub b/ for maximum-likelihood decoding of binary linear block codes is investigated. The contribution P/sub b/(j) of each information bit j to P/sub b/ is considered and an upper bound on P/sub b/(j) is derived. For randomly generated codes, it is shown that the conventional approximation at high SNR P/sub b//spl ap/(d/sub H//N).P/sub s/, where P/sub s/ represents the block error probability, holds for systematic encoding only. Also systematic encoding provides the minimum P/sub b/ when the inverse mapping corresponding to the generator matrix of the code is used to retrieve the information sequence. The bit-error performances corresponding to other generator matrix forms are also evaluated. Although derived for codes with a generator matrix randomly generated, these results are shown to provide good approximations for codes used in practice. Finally, for soft-decision decoding methods which require a generator matrix with a particular structure such as trellis decoding, multistage decoding, or algebraic-based soft-decision decoding, equivalent schemes that reduce the bit-error probability are discussed. Although the gains achieved at practical bit-error rates are only a fraction of a decibel, they remain meaningful as they are of the same orders as the error performance differences between optimum and suboptimum decodings. Most importantly, these gains are free as they are achieved with no or little additional circuitry which is transparent to the conventional implementation.

In this correspondence, various aspects of reliability-based syndrome decoding of binary codes are investigated. First, it is shown that the least reliable basis (LRB) and the most reliable basis (MRB) are dual of each other. By exploiting this duality, an algorithm performing maximum-likelihood (ML) soft-decision syndrome decoding based on the LRB is presented. Contrarily to previous LRB-based ML syndrome decoding algorithms, this algorithm is more conveniently implementable for codes whose codimension is not small. New sufficient conditions for optimality are derived. These conditions exploit both the ordering associated with the LRB and the structure of the code considered. With respect to MRR-based sufficient conditions, they present the advantage of requiring no soft information and thus can be preprocessed and stored. Based on these conditions, low-complexity soft-decision syndrome decoding algorithms for particular classes of codes are proposed. Finally, a simple algorithm is analyzed. After the construction of the LRB, this algorithm computes the syndrome of smallest Hamming weight among o(K/sup i/) candidates, where K is the dimension of the code, for an order i of reprocessing. At practical bit-error rates, for codes of length N/spl les/128, this algorithm always outperforms any algebraic decoding algorithm capable of correcting up to t+1 errors with an order of reprocessing of at most 2, where t is the error-correcting capability of the code considered.

This paper presents an efficient trellis-based maximum-likelihood decoding algorithm for binary linear block codes. This algorithm is recursive in nature and is devised based on the structural properties and optimum sectionalization of a code trellis. The complexity of the proposed decoding algorithm is analyzed. Numerical results show that the proposed decoding algorithm significantly reduces the decoding complexity. A recursive method for finding the optimum sectionalization of a trellis in terms of computational complexity is given.

The soft decision decoding algorithm based on the ordered statistics proposed by Fossorier and Lin (see IEEE Trans. Inform. Theory, vol.41, no.9, p.1379-96, 1995) is applied to the Rayleigh fading channel with coherent detection. For an (N, K) block code, it is shown that order-1 reprocessing, or equivalently considering K+1 codeword candidates, provides most of the coding gain over uncoded binary phase shift keying (BPSK). In addition to its contribution to coding for the Rayleigh fading channel, the article also provides a general framework for evaluating the error performance of an algorithm based on a total or partial ordering of a random variable (RV) depending on one or many other RVs and illustrates how the reprocessing method of Fossorier et al. relates to the reliability measures defining the ordering.

This letter generalizes the coset decoding of decomposable codes, offering further refinements in the tradeoffs among error performance, decoding complexity, and decoding speed.

This paper presents a method for evaluating the bit-error probability of a concatenated coding system for BPSK transmission over the AWGN channel. In the concatenated system, a linear binary block code is used as the inner code and is decoded with the soft-decision maximum likelihood decoding, and a maximum distance separable code (or its interleaved code) is used as the outer code and is decoded with a bounded distance decoding. The method is illustrated through a specific example in which the inner code is a binary (64.40.8) Reed-Muller subcode and the outer code is the NASA standard (255, 223, 33) Reed-Solomon code over GF(2/sup 8/) interleaved to a depth of 5. This specific concatenated system is being considered for NASA's high-speed satellite communications. The bit-error performance is evaluated by a combination of simulation and analysis. The split weight enumerators for the maximum distance separable codes are derived and used for the analysis.

This paper investigates trellis structures of linear block codes for the integrated circuit (IC) implementation of Viterbi decoders capable of achieving high decoding speed while satisfying a constraint on the structural complexity of the trellis in terms of the maximum number of states at any particular depth. Only uniform sectionalizations of the code trellis diagram are considered. An upper-bound on the number of parallel and structurally identical (or isomorphic) subtrellises in a proper trellis for a code without exceeding the maximum state complexity of the minimal trellis of the code is first derived. Parallel structures of trellises with various section lengths for binary BCH and Reed-Muller (RM) codes of lengths 32 and 64 are analyzed. Next, the complexity of the IC implementation of a Viterbi decoder based on an L-section trellis diagram for a code is investigated. A structural property of a Viterbi decoder called add-compare-select (ACS)-connectivity which is related to state connectivity is introduced. This parameter affects the complexity of wire-routing (interconnections within the IC). The effect of five parameters namely: (1) effective computational complexity; (2) complexity of the ACS-circuit; (3) traceback complexity; (4) ACS-connectivity; and (5) branch complexity of a trellis diagram on the very large scale integration (VLSI) complexity of a Viterbi decoder is investigated. It is shown that an IC implementation of a Viterbi decoder based on a nonminimal trellis requires less area and is capable of operation at higher speed than one based on the minimal trellis when the commonly used ACS-array architecture is considered.

This letter presents a technique of combining multilevel coded modulation and product coding to form product modulation codes which achieve low bit error rates with reduced decoding complexity. Three multistage decoding algorithms are presented, and good codes for both the additive white Gaussian noise (AWGN) and Rayleigh fading channels have been constructed.

The first part of this paper presents a simple and systematic technique for constructing multidimensional M-ary phase shift keying (MPSK) trellis coded modulation (TCM) codes. The construction is based on a multilevel concatenation approach. In which binary convolutional codes with good free branch distances are used as the outer codes and block MPSK modulation codes are used as the inner codes (or the signal spaces). Conditions on phase invariance of these codes are derived and a multistage decoding scheme for these codes is proposed. The proposed technique can be used to construct good codes for both the additive white Gaussian noise (AWGN) and fading channels as is shown in the second part of this paper.

For pt.I see ibid., vol.45, no.1, p.64-72, 1997. We use the construction technique proposed in part I to construct multidimensional trellis coded modulation (TCM) codes for both the additive white Gaussian noise (AWGN) and the fading channels. Analytical performance bounds and simulation results show that these codes perform very well and achieve significant coding gains over uncoded reference modulation systems. In addition, the proposed technique can be used to construct codes which have a performance/decoding complexity advantage over the codes listed in literature.

In this correspondence, the exact weight distribution for closest coset decoding of |u|u+v| constructed codes is derived. The results allow more accurate evaluations of the decoding error probabilities.

This correspondence presents a hybrid reliability-based decoding algorithm which combines the reprocessing method based on the most reliable basis and a generalized Chase-type algebraic decoder based on the least reliable positions. It is shown that reprocessing with a simple additional algebraic decoding effort achieves significant coding gain. For long codes, the order of reprocessing required to achieve asymptotic optimum error performance is reduced by approximately 1/3. This significantly reduces the computational complexity, especially for long codes. Also, a more efficient criterion for stopping the decoding process is derived based on the knowledge of the algebraic decoding solution.

Codes with decomposable structure allow the use of multistage decoding procedures to achieve suboptimum bounded-distance error performance with reduced decoding complexity. This correspondence presents some new decomposable codes, including a class of distance-8 codes, that are constructed based on the |a+x|b+x|a+b+x| construction method. Some existing best codes are shown to be decomposable and hence can be decoded with multistage decoding.

This article presents a suboptimum soft-decision decoding scheme for binary linear block codes based on an iterative search algorithm. The scheme uses an algebraic decoder to iteratively generate a sequence of candidate codewords one at a time using a set of test error patterns that are constructed based on the reliability information of the received symbols. When a candidate codeword is generated, it is tested based on an optimality condition. If it satisfies the optimality condition, then it is the most likely (ML) codeword and the decoding stops. If it fails the optimality test, a search for the ML codeword is conducted in a region which contains the ML codeword. The search region is determined by the current candidate codeword and the reliability of the received symbols. The search is conducted through a purged trellis diagram for the given code using the Viterbi algorithm. If the search fails to find the ML codeword, a new candidate is generated using a new test error pattern, and the optimality test and search are renewed. The process of testing and search continues until either the ML codeword is found or all the test error patterns are exhausted and the decoding process is terminated. Numerical results show that the proposed decoding scheme achieves either practically optimal performance or a performance only a fraction of a decibel away from the optimal maximum-likelihood decoding with a significant reduction in decoding complexity compared with the Viterbi decoding based on the full trellis diagram of the codes.

Coset codes are considered as terminated convolutional codes. Based on this approach, three new general results are presented. First, it is shown that the iterative squaring construction can equivalently be defined from a convolutional code whose trellis terminates. This convolutional code determines a simple encoder for the coset code considered, and the state and branch labelings of the associated trellis diagram become straightforward. Also, from the generator matrix of the code in its convolutional code form, much information about the trade-off between the state connectivity and complexity at each section, and the parallel structure of the trellis, is directly available. Based on this generator matrix, it is shown that the parallel branches in the trellis diagram of the convolutional code represent the same coset code C/sub 1/ of smaller dimension and shorter length. Utilizing this fact, a two-stage optimum trellis decoding method is devised. The first stage decodes C/sub 1/ while the second stage decodes the associated convolutional code, using the branch metrics delivered by stage 1. Finally, a bidirectional decoding of each received block starting at both ends is presented. If about the same number of computations is required, this approach remains very attractive from a practical point of view as it roughly doubles the decoding speed. This fact is particularly interesting whenever the second half of the trellis is the mirror image of the first half, since the same decoder can be implemented for both parts.

The error performance of a modulation code over a channel depends on several distance parameters and the path multiplicity of the code. For the AWGN channel, the error performance of a modulation code depends mainly on its minimum squared Euclidean distance and path multiplicity. For the Rayleigh fading channel, however, the error performance of a modulation code depends strongly on its minimum symbol distance, minimum product distance, and path multiplicity. It depends on the minimum squared Euclidean distance in a lesser degree. This paper is concerned with the construction of block and trellis MPSK modulation codes for the Rayleigh fading channel. In each construction, the distance parameters are chosen to achieve good error performance with reduced decoding complexity.

Soft-decision decoding of a linear block code using the most reliable basis corresponding to each received word is investigated. Based either on probabilistic properties or on the structure of the code considered, three improvements to the algorithm devised by Fossorier and Lin (see ibid., vol.41, no.9, p.1379-1396, 1995) are presented. These modifications allow large computation savings or significant decoding speedup with little error performance degradation. First, a reduced probabilistic list of codeword candidates is associated with order-i reprocessing of a given code. It results in a large reduction of the maximum number of computations with a very small degradation in performance. Then, a probabilistic stopping criterion is introduced for order-0 reprocessing. This new test significantly decreases the average number of computations when appropriately implemented. Finally, the application of the algorithm to coset decoding is considered for |u|u+v| constructed codes. In addition to the conventional coset decoding, a new adaptive practically optimum coset decoding method is presented where at each reprocessing stage, the number of surviving cosets decreases. Suboptimum closest coset decoding is also investigated. It is shown that two-stage decoding with the algorithm of Fossorier and Lin offers a large variety of choices, since the reprocessing order of each stage can be determined independently.

Soft-decision decoding algorithms of binary linear block codes require reordering of the received symbols within each block in decreasing reliability. Efficient decoding algorithms based on reordering have been devised. This paper presents different results related to the ordering of a sequence of N received symbols with respect to their reliability measure, for BPSK transmission over the AWGN channel model. First, a tight approximation of Pe(i; N), the probability that the hard decision associated with the ith symbol of the ordered sequence is in error, is derived. Then, it is shown that despite the fact that the random variables representing the noise at positions n/sub 1/, n/sub 2/,...,n/sub j/ of the ordering are no longer independent, the events of having a hard decision decoding error at these positions remain almost independent. Pe(n/sub 1/,n/sub 2/,...,n/sub j/; N), the probability that the hard decisions associated with the symbols at positions n/sub 1/, n/sub 2/,...n/sub j/ in the ordered sequence are in error, is thus well approximated from each of the Pe (n/sub i/; N), for i/spl isin/[1,j]. Finally, based on the independence of these events, the fully connected 2/sup N/-state BSC representing the channel after ordering is simplified by N independent time-shared 2-state BSCss. This new model allows one to easily and tightly approximate the capacity of the channel after ordering.

This paper introduces new block coded 8-PSK modulations with unequal error protection (UEP) capabilities for Rayleigh fading channels. The design of efficient block coded modulations (BCM) over 8-PSK signal sets, for the specific purpose of UEP, over Rayleigh fading channels is considered. UEP is desirable in communications systems where part of the source information is more important, or error sensitive, such as the transmission of coded speech and data broadcasting. The proposed block modulation codes are based on the multilevel construction of Imai and Hirakawa (1977). It is shown that the use of binary linear UEP (LUEP) codes as component codes in one or two of the encoding levels provides, in addition to superior UEP capabilities, a higher error performance, at the expense of a very modest reduction in bandwidth efficiency, with respect to conventional multilevel codes. Computer simulation results show that, over a Rayleigh fading channel, a significant improvement in the coding gain is obtained by the use of binary LUEP codes as constituent codes in the multilevel construction.

The paper shows that the catastrophic behavior of TCM codes based on partially overlapped signal constellations significantly increases both the effective error coefficient and the decoding delay of such codes, resulting in a non negligible performance degradation with respect to the asymptotic coding gain.>

Nested concatenated coded 8-PSK modulation schemes are proposed to achieve good error performance, large coding gains, and high spectral efficiency with reduced decoding complexity. In the proposed schemes, Reed-Solomon (RS) codes, including the NASA standard (255, 223) RS code, are used as the outer codes and multidimensional trellis 8-PSK codes are used as the inner codes. The inner codes are constructed from convolutional codes with good free branch distances and multidimensional 8-PSK coset codes through concatenation. These schemes are analyzed and upper bounds on their error performances are derived. Analytical and simulation results show that these schemes achieve large coding gains over uncoded reference systems for bit-error-rates below 10/sup -3/, with reduced decoding complexity.>

Presents a novel approach to soft decision decoding for binary linear block codes. The basic idea is to achieve a desired error performance progressively in a number of stages. For each decoding stage, the error performance is tightly bounded and the decoding is terminated at the stage where either near-optimum error performance or a desired level of error performance is achieved. As a result, more flexibility in the tradeoff between performance and decoding complexity is provided. The decoding is based on the reordering of the received symbols according to their reliability measure. The statistics of the noise after ordering are evaluated. Based on these statistics, two monotonic properties which dictate the reprocessing strategy are derived. Each codeword is decoded in two steps: (1) hard-decision decoding based on reliability information and (2) reprocessing of the hard-decision-decoded codeword in successive stages until the desired performance is achieved. The reprocessing is based on the monotonic properties of the ordering and is carried out using a cost function. A new resource test tightly related to the reprocessing strategy is introduced to reduce the number of computations at each reprocessing stage. For short codes of lengths N/spl les/32 or medium codes with 32>

Unequal error protection (UEP) codes find applications in broadcast channels, as well as in other digital communication systems, where messages have different degrees of importance. Binary linear UEP (LUEP) codes combined with a Gray mapped QPSK signal set are used to obtain new efficient QPSK block-modulation codes for unequal error protection. Several examples of QPSK modulation codes that have the same minimum squared Euclidean distance as the best QPSK modulation codes, of the same rate and length, are given. In the new constructions of QPSK block-modulation codes, even-length binary LUEP codes are used. Good even-length binary LUEP codes are obtained when shorter binary linear codes are combined using either the well-known |u~|u~+v~|-construction or the so-called construction X. Both constructions have the advantage of resulting in optimal or near-optimal binary LUEP codes of short to moderate lengths, using very simple linear codes, and may be used as constituent codes in the new constructions. LUEP codes lend themselves quite naturally to multistage decoding up to their minimum distance, using the decoding of component subcodes. A new suboptimal two-stage soft-decision decoding of LUEP codes is presented and its application to QPSK block-modulation codes for UEP illustrated.>

Presents a class of binary primitive BCH codes that have unequal-error-protection (UEP) capabilities. The authors use a previous result on the span of their minimum weight vectors to show that binary primitive BCH codes, containing second-order punctured Reed-Muller (RM) codes of the same minimum distance, are binary-cyclic UEP codes. The values of the error correction levels for this class of binary LUEP codes are estimated.>

A class of block QPSK modulation codes for unequal error protection (UEP) is presented. These codes are particularly suitable either for broadcast channels or for communication systems where parts of the information messages are more important than others. An example of the latter is coded speech transmission. Not much is known on the application of block UEP codes in combined coding and modulation schemes. We exhibit a method to combine binary linear UEP (LUEP) block codes of even length, using a Gray mapping, with a QPSK signal set to construct efficient block QPSK modulation codes with nonuniform error protection capabilities for bandwidth efficient transmission over AWGN (additive white Gaussian noise) and Rayleigh fading channels.

An upper bound on the effective error coefficient of a two-level code with two-stage decoding is presented. This bound provides a guideline for constructing two-level codes to achieve a good trade-off between the error performance and decoding complexity. Based on this bound, good two-level decompositions of some Reed-Muller codes for two-stage decoding are found. Simulation results on the error performances of some Reed-Muller codes of lengths up to 64 with two-stage soft-decision suboptimum decoding based on their two-level decompositions are given.>

Several authors have addressed the problem of designing good linear unequal error protection (LUEP) codes. However, very little is known about good nonbinary LUEP codes. The authors present a class of optimal nonbinary LUEP codes for two different sets of messages. By combining t-error-correcting Reed-Solomon (RS) codes and shortened nonbinary Hamming codes, they obtain nonbinary LUEP codes that protect one set of messages against any t or fewer symbol errors and the remaining set of messages against any single symbol error. For t/spl ges/2, they show that these codes are optimal in the sense of achieving the Hamming lower bound on the number of redundant symbols of a nonbinary LUEP code with the same parameters.>

To decode a long block code with a large minimum distance by maximum likelihood decoding is practically impossible because the decoding complexity is simply enormous. However, if a code can be decomposed into constituent codes with smaller dimensions and simpler structure, it is possible to devise a practical and yet efficient scheme to decode the code. This paper investigates a class of decomposable codes, their distance and structural properties. It is shown that this class includes several classes of well-known and efficient codes as subclasses. Several methods for constructing decomposable codes or decomposing codes are presented. A two-stage (soft-decision or hard-decision) decoding scheme for decomposable codes, their translates or unions of translates is devised, and its error performance is analyzed for an AWGN channel. The two-stage soft-decision decoding is suboptimum. Error performances of some specific decomposable codes based on the proposed two-stage soft-decision decoding are evaluated. It is shown that the proposed two-stage suboptimum decoding scheme provides an excellent trade-off between the error performance and decoding complexity for codes of moderate and long block length.>

The multilevel coding technique is used for constructing multilevel trellis M-ary phase-shift-keying (MPSK) modulation codes for the Rayleigh fading channel. In the construction of a code, all the factors which affect the code performance and its decoding complexity are considered. The error performance of some of these codes based on both one-stage optimum decoding and multistage suboptimum decoding has been simulated. The simulation results show that these codes achieve good error performance with small decoding complexity.>

It was shown earlier that for a punctured Reed-Muller (RM) code or a primitive BCH code, which contains a punctured RM code of the same minimum distance as a large subcode, the state complexity of the minimal trellis diagrams is much greater than that for an equivalent code obtained by a proper permutation of the bit positions. The problem of finding a permutation of the bit positions for a given code that minimizes the state complexity of its minimal trellis diagram is related to the generalized Hamming weight hierarchy of a code, and it is shown that, for RM codes, the standard binary order of bit positions is optimum at every bit position with respect to the state complexity of a minimal trellis diagram by using a theorem due to V.K. Wei (1991). The state complexity of the trellis diagram for the extended and permuted (64, 24) BCH code is discussed.>

An upper bound on the number of states of a minimal trellis diagram for a linear block code is derived. Using this derivation a cyclic (or shortened cyclic) code or its extended code is shown to be the worst in terms of trellis state complexity among the linear codes of the same length and dimension. The complexity of the minimal trellis diagrams for linear block codes of length 2/sup m/, including the Reed-Muller codes, is analyzed. The construction of minimal trellis diagrams for some extended and permuted primitive BCH codes is presented. It is shown that these codes have considerably simpler trellis structure than the original codes in cyclic form without bit-position permutation.>

Multistage decoding of multilevel block multilevel phase-shift keying (M-PSK) modulation codes for the additive white Gaussian noise (AWGN) channel is investigated. Several types of multistage decoding, including a suboptimum soft-decision decoding scheme, are devised and analyzed. Upper bounds on the probability of an incorrect decoding of a code are derived for the proposed multistage decoding schemes. Error probabilities of some specific multilevel block 8-PSK modulation codes are evaluated and simulated. The computation and simulation results for these codes show that with multistage decoding, significant coding gains can be achieved with large reduction in decoding complexity. In one example, it is shown that the difference in performance between the proposed suboptimum multistage soft-decision decoding and the single-stage optimum decoding is small, only a fraction of a dB loss in SNR at the block error probability of 10/sup -6/.>

The authors propose retransmission error control schemes for broadcast channels. In their selective-repeat-ARQ (automatic repeat request) method, they incorporate some aspects of implementation that they believe were ignored by all the previous work. Based on this protocol, they also propose a type-2 hybrid ARQ scheme that uses parity retransmission. They apply the dynamic programming optimization technique of Wang and Silvester (1987), with some modification, to their schemes. Analysis shows that their schemes outperform all the existing schemes. As special cases of their proposed schemes, the authors can obtain point-to-point ARQ schemes which also outperform all the existing point-to-point schemes. Thus, their schemes extend the useful range of retransmission error control schemes for these channels.>

New combinatorial and algebraic techniques are presented for systematically different (d,k) block codes capable of detecting and correcting single bit-errors, single-peak shift-errors, double adjacent-errors and multiple adjacent erasures. Constructions utilizing channel side information, such as the magnetic recording ternary channel output string, or erasures, do not impose any restriction on The k-constraint, while some of the other constructions require k=2d. Due to the small and fixed number of redundant bits, the rates of both classes of constructions can be made to approach the capacity of the d-constrained channel for long codeword lengths. All the codes can be encoded and decoded with simple, structured logic circuits.>

Two important structural properties of block M(=2/sup '/)-ary PSK modulation codes, linear structure and phase symmetry, are investigated. An M-ary modulation code is first represented as a code with symbols from the integer group S/sub M-PSK/=(0,1,2,---,M-1) under modulo-M addition. Then the linear structure of block M-PSK modulation codes over S/sub M-PSK/ with respect to modulo-M vector addition is defined, and conditions are derived under which a block M-PSK modulation code is linear. Once the linear structure is developed, the phase symmetry of block M-PSK modulation codes is studied. In particular, a necessary and sufficient condition for a block M-PSK modulation code that is linear as a binary code to be invariant under 2/sup h/180 degrees /M phase rotation, for 1>

The multilevel technique for combining block coding and modulation is investigated. A general formulation is presented for multilevel modulation codes in terms of component codes with appropriate distance measures. A specific method for constructing multilevel block modulation codes with interdependency among component codes is proposed. Given a multilevel block modulation code C with no interdependency among the binary component codes, the proposed method gives a multilevel block modulation code C' that has the same rate as C, a minimum squared Euclidean distance not less than that of C, a trellis diagram with the same number of states as that of C, and a smaller number of nearest neighbor codewords than that of C. Finally, a technique is presented for analyzing the error performance of block modulation codes for an additive white Gaussian noise (AWGN) channel based on soft-decision maximum likelihood decoding. Error probabilities of some specific codes are evaluated by simulation and upper bounds based on their Euclidean weight distributions.>

A concatenated coded modulation scheme is presented for error control in data communications. The scheme is achieved by concatenating a Reed-Solomon outer code and a bandwidth efficient block inner code for M-ary phase-shift keying (PSK) modulation. Error performance of the scheme is analyzed for an additive white Gaussian noise (AWGN) channel. It is shown that extremely high reliability can be attained by using a simple M-ary PSK modulation inner-code and a relatively powerful Reed-Solomon outer code. Furthermore, if an inner code of high effective rate is used, the bandwidth expansion required by the scheme due to coding will be greatly reduced. The scheme is particularly effective for high-speed satellite communications for large file transfer where high reliability is required. A simple method is also presented for constructing block codes for M-ary PSK modulation. Soome short M-ary PSK codes with good minimum squared Euclidean distance are constructed. These codes have trellis structure and hence can be decoded with a soft-decision Viterbi decoding algorithm. Furthermore, some of these codes are phase invariant under multiples of 45 degrees rotation.>

A robust error control coding system is presented. This system is a cascaded FEC (forward error control) scheme supported by parity retransmissions for further error correction in the erroneous data words. The error performance and throughput efficiency of the system are analyzed. Two specific examples of the error control system are studied. The first example does not use an inner code, and the outer code, which is not interleaved, is a shortened code of the NASA standard RS code over GF(2/sup 8/). The second example, as proposed for NASA uses the same shortened RS code as the base outer code C/sub 2/, except that it is interleaved to a depth of 2. It is shown that both examples provide high reliability and throughput efficiency even for high channel bit-error rates in the range of 10/sup -2/.>

Exhaustive computation by a computer was used to find the unequal error protection capabilities of all binary cyclic codes of odd length up to 65 that have minimum distances of at least 3. For those codes for which upper bounds can only be computed on their unequal error protection capabilities, an analytic method developed by V.N. Dynkin and V.A. Togonidze (1976) is used to show that the upper bounds meet the exact unequal error protection capabilities.>

Investigates the error detecting capabilities of the shortened hamming codes adopted for error detection in IEEE Standard 802.3. These codes are also used for error detection in the data link layer of the Ethernet, a local area network. The authors compute the weight distributions for various code lengths. From the results, they show the probability of undetectable error and that of detectable error for a binary symmetric channel with bit-error rate 10/sup -5/>

A coding scheme for error control in data communication systems is investigated. The scheme is obtained by cascading two error-correcting codes, called the inner and outer codes. Its error performance is analyzed for a binary symmetric channel with a bit-error rate epsilon>

The unequal error correction capabilities of binary cyclic codes of composite length are investigated. Under certain conditions, direct sums of concatenated codes have unequal error correction capabilities. By a modified Hartmann and Tzeng (1973) algorithm, it is shown that a binary cyclic code of composite length is equivalent to the direct sum of concatenated codes. With this, some binary cyclic unequal error protection (UEP) codes are constructed. Finally, the authors present a class of two-level UEP cyclic direct-sum codes which provide error correction capabilities higher than those guaranteed by the Blokh-Zyablov (1974) constructions.>

In this paper, a concatenated coding scheme for error control in data communications is presented and analyzed. In this scheme, the inner code is used for both error correction and detection; however, the outer code is used only for error detection. A retransmission is requested if either the inner code decoder fails to make a successful decoding or the outer code decoder detects the presence of errors after the inner code decoding. Probability of undetected error (or decoding error) of the proposed scheme is derived. An efficient method for computing this probability is presented. Throughput efficiency of the proposed error control scheme incorporated with a selective-repeat ARQ retransmission strategy is also analyzed. Three specific examples are presented. One of the examples is proposed for error control in the NASA Telecommand System.

Recently Kasami {\em et al.} presented a linear programming approach to the weight distribution of binary linear codes [2]. Their approach to compute upper and lower bounds on the weight distribution of binary primitive BCH codes of length2^{m} - 1withm \geq 8and designed distance2t + 1with4 \leq t \leq 5is improved. From these results, the relative deviation of the number of codewords of weightj\leq 2^{m-1}from the binomial distribution2^{-mt} \left( \stackrel{2^{m}-1}{j} \right)is shown to be less than 1 percent for the following cases: (1)t = 4, j \geq 2t + 1andm \geq 16; (2)t = 4, j \geq 2t + 3and10 \leq m \leq 15; (3)t=4, j \geq 2t+5and8 \leq m \leq 9; (4)t=5,j \geq 2t+ 1andm \geq 20; (5)t=5, j \geq 2t+ 3and12 \leq m \leq 19; (6)t=5, j \geq 2t+ 5and10 \leq m \leq 11; (7)t=5, j \geq 2t + 7andm=9; (8)t= 5, j \geq 2t+ 9andm = 8.

A trellis code is {\em homogeneous} if the number of branches emanating from each node (or state) in the trellis diagram is constant. For example, convolutional codes are linear homogeneous trellis codes. A trellis code is {\em nonhomogeneous} if the number of branches emanating from each node in the trellis diagram is not the same. The two-user binary adder channel is a multiple-access channel with two binary inputs,x_{1}andx_{2}, and one ternary output,y = x_{1} + x_{2}, where the addition is done in the real number field. The adder channel is synchronous if both encoders and the decoder maintain block (frame) synchronism. It is quasi-synchronous if the encoders do not start their blocks at the same time, but the decoder knows the position of each block. The difference between the starting times of the blocks is called the slippage. The channel is asynchronous if no block synchronism exists among the encoders and the decoder. Some uniquely decodable code pairs(C_{1}, C_{2})are presented that can be used to transmit information reliably over the quasi-synchronous binary adder channel with two users. One of the codes is a nonhomogeneous trellis code, the other is a common block code. Our code rates are better than Deaett-Wolf codes and are close to or equal to the asymptotic rates of Kasami {\em et al}. A method for calculating the rates of nonhomogeneous trellis codes is described. An algorithm for finding more uniquely decodable code pairs for the quasi-synchronous binary adder channel is formulated.

Shortened Hamming codes are widely used for error detection in data communications. In this paper, a method for computing the probability of an undetected error for these codes is presented. This method is then used to evaluate the error-detection performance of the shortened codes obtained from the two distance-4 Hamming codes adopted by CCITT X.25 for error control for packet-switched networks. We show that shortening a code does affect its error-detection performance.

Binary primitive BCH codes form a large class of powerful error-correcting codes. The weight distributions of primitive BCH codes are unknown except for some special classes, such as the single, double, triple error-correcting codes and some very low-rate primitive BCH codes. However, asymptotic results for the weight distribution of a large subclass of primitive BCH codes have been derived by Sidel'nikov. These results provide some insight into the weight structure of primitive BCH codes. Sidel'nikov's approach is improved and applied to the weight distribution of any binary linear block code. Then Sidel'nikov's results on the weight distributions of binary primitive BCH codes are improved and it is shown that the weights of a binary primitive code have approximate binomial distribution.

Block coding for the binary symmetric broadcast channel with two receivers is investigated. A graph-theoretic approach to the construction of a class of block codes with unequal error protection for two different sets of messages is presented. A code in this class is a direct sum of two component codes; each set of messages is encoded based on one component code. The codes in this class are easy to implement. Decoding of these codes is presented, and lower bounds on the achievable rates of these codes are derived. The bounds are tighter than the Katsman's bounds.

In this paper we investigate the performance of maximum-distance-separable codes with symbols fromGF(q)when they are used for pure error detection or for simultaneous error correction and detection over aq-input andq-output discret memoryless channel with symbol error probability ε. First we show that the probability of undetected error for an MDS code used for pure error detection is upper bounded byq^{-r}and decreases monotonically as εdecreases from(q - 1)/qto 0, whereris the number of parity-check symbols of the code. Then we show that the probability of undetected error for an MDS code used for correctingtor fewer symbol errors is upper bounded byq^{-r} \Sum\min{i=0}\max{t}(\min{i} \max{n})(q - 1)^{i}and decreases monotonically as ε decreases from(q - 1)/qto 0. These results show that the MDS codes are effective for both pure error detection and simultaneous error correction and detection.

The hybrid ARQ scheme with parity retransmission for error control, recently proposed by Lin and Yu [1], [2], is quite robust. This scheme provides both high system throughput and high system reliability. In this paper, a modified Lin-Yu hybrid ARQ scheme is presented. The modified scheme provides a slightly better throughput performance than the original Lin-Yu scheme; however, it is more flexible in utilizing the error-correction power of a code. The modified scheme can be incorporated with a rate 1/2 convolutional code using Viterbi decoding. Furthermore, the pure selectiverepeat ARQ is a degenerated case of the modified scheme in selective mode. Lin and Yu analyzed their scheme only for a receiver buffer of sizeNwhereNis the number of data blocks that can be transmitted in a round-trip delay interval. No analysis for other buffer sizes was given. In this paper, the throughput performance of the modified Lin-Yu scheme is analyzed for any size of receiver buffer. Consequently, the throughput efficiency of the pure selective-repeat ARQ for any receiver buffer size can be obtained. We also show that the modified scheme achieves the same order of reliability as a pure ARQ scheme.

The probability of undetected error of linear block codes for use on a binary symmetric channel is investigated. Upper hounds are derived. Several classes of linear block codes are proved to have good error-detecting capability.

We relate coding for the two-user multiple-access binary adder channel to a problem in graph theory, known as the independent set problem. Graph-theoretic approaches to coding for both synchronized and nonsynchronized two-user adder channels are presented. Using the Tuŕan theorem on the independence number of a simple graph, we are able to improve the lower bounds on the achievable rates of uniquely and\delta-decodable codes for the synchronized adder channel derived by Kasami and Lin. We are also able to derive lower bounds on the achievable rates of uniquely decodable codes for the nonsynchronized adder channel. We show that the rates of Deaett-Wolf codes for the nonsynchronized adder channel fall below the bounds. Synchronizing sequences for the nonsynchronized adder channel are constructed.

This paper presents a new type of hybrid ARQ scheme for error control in data communication systems. The new scheme is based on the concept that the parity-check digits for error correction are sent to the receiver only when they are needed. Normally, data blocks with some parity-check bits for error detection are transmitted. When a data blockDis detected in errors, the retransmissions are not simply repetitions ofD, but alternate repetitions of a parity blockP(D)andD. The parity blockP(D)is formed based onDand a half-rate invertible code which is capable of correctingtor fewer errors and simultaneously detectingd (d > t)or fewer errors. When a parity block is received, it is used to recover the originally transmitted data block either by inversion or by decoding operation. The repetitions of the parity blockP(D)and the data blockDare alternately stored in the receiver buffer for error correction untilDis recovered. We show that the proposed hybrid ARQ scheme provides both high system throughput and high system reliability. It is particularly attractive for error control in high-speed data communication systems with significant roundtrip delays, such as satellite channels.

In high bit rate data transmission systems with ARQ error control, the throughput efficiency is a function of bit error rate, block or packet size, and the effect of significant round trip delays such as may be experienced in satellite communication systems. The selective-repeat ARQ scheme is capable of providing superior throughput performance independent of round trip delay, but requires excessively large receiver buffers; as a result the inferior GoBackNprocedure is commonly adopted. This paper analyzes a class of mixed-mode ARQ protocol models which incorporate a selectiverepeat mode with finite receiver buffer. The protocol models are shown to be amenable to exact throughput analysis, but do assume that the round trip delay is constant and known, blocks are of constant length, and the ACK/NAK signals are returned error free. These assumptions might create difficulties for practical implementation. However, the analytical model results highlight those aspects of ARQ protocols which affect throughput performance as round trip delays increase. The results show that it is desirable for best throughput performance in practical systems that at least the first retransmission of a block following an error should be in the selective-repeat mode to obtain superior performance over GoBackNschemes. Furthermore, alternative secondary retransmission modes are considered which ensure that reliable transmission can be achieved without receiver buffer overflow, even if the selective-repeat mode retransmissions fail. It is shown that the choice of secondary mode does not have a significant effect on the throughput efficiency but has a bearing on complexity.

In this paper, we investigate a selective-repeat ARQ scheme which operates with a finite receiver buffer and a finite range of sequence numbers. The throughput performance of the proposed scheme is analyzed and simulated based on the assumption that the channel errors are randomly distributed and the return channel is noiseless. Both analytical and simulation results show that it significantly outperforms the go-back-NARQ scheme, particularly for channels with large roundtrip delay and high data rate. It provides high throughput efficiency over a wide range of bit error rates. The throughput remains in a usable range even for very high error rate conditions. The proposed scheme is capable of handling data and/or acknowledgment loss. Furthermore, when buffer overflow occurs at the receiver, the transmitter is capable of detecting it and backs up to the proper location of the input queue to retransmit the correct data blocks.

The conventional go-back-NARQ is inadequate for error control on satellite channels due to the large round-trip delay and high bit rate. The throughput efficiency of this system drops rapidly as the channel error rate increases. In this paper, a variation of the go-back-NARQ is described. This variation reduces the effect of the round-trip delay, and hence increases the system throughput efficiency. Therefore, it may find applications m satellite communication systems or other systems where round-trip delay is large.

Block coding for a memoryless two-input single-output multiple-access channel, called a two-user adder channel, is studied. Techniques for constructing codes for this particular channel are presented. Upper and lower bounds on the achievable rates of these codes are derived. These bounds define various two-dimensional regions for the achievable rates of codes for the two-user adder channel.

A scheme is presented for decoding linear\delta-decodable codes for the two-user noisy adder channel that exploits the linearity of the codes and corrects all patterns of\lfloor (\delta - 1)/2 \rflooror fewer transmission errors.

In this paper, coding for a multiple-access discrete memoryless channel is investigated. Block codes which are uniquely decodable and capable of correcting errors are constructed.

This paper presents a class of majority-logic decodable codes whose structure is based on the structural properties of Euclidean geometries (EG) and codes that are invariant under the affine group of permutations. This new class of codes contains the ordinary EG codes and some generalized EG codes as subclasses. One subclass of new codes is particularly interesting: they are the most efficient majority-logic decodable codes that have been constructed.

For1 \leq i \leq m - s- 2and0 \leq s \leq m -2i, the intersection of the binary BCH code of designed distance2 ^{m-s-1} - 2 ^{m-s-t-1} - 1and length2^m - 1with the shortened(s + 2)th-order Reed-Muller code of length2^m -- 1has codewords of weight2^{m-s-1} - 2^{m-s-t-1} - 1.

A method of shortening finite analytic geometry codes, projective-geometry (PG) codes, Euclidean-geometry (EG) codes, and 2-fold EG codes is presented. The shortened codes preserve the feature of being majority-logic decodable and they have the same error-correcting capability as the original codes. Combinatorial expressions for the parity-check symbols of the shortened codes are derived.

Polynomial codes and their dual codes as introduced by Kasami, Lin, and Peterson have considerable algebraic and geometric structure. It has been shown that these codes contain many well-known classes of cyclic codes as subclasses, such as BCH codes, projective geometry codes (PG codes), Euclidean geometry codes (EG codes), and generalized Reed-Muller codes (GRM codes). In this paper, combinatorial expressions for the number of information symbols and parity-check symbols in polynomial codes are derived. The results are applied to two important subclasses of codes, the PG codes and EG codes.

The class of polynomial codes introduced by Kasami et al. has considerable inherent algebraic and geometric structure. It has been shown that this class of codes and their dual codes contain many important classes of cyclic codes as subclasses, such as BCH codes, Reed-Solomon codes, generalized Reed-Muller codes, projective geometry codes, and Euclidean geometry codes. The purpose of this paper is to investigate further properties of polynomial codes and their duals. First, majority-logic decoding for the duals of certain primitive polynomial codes is considered. Two methods of forming nonorthogonal parity-check sums are presented. Second, the maximality of Euclidean geometry codes is proved. The roots of the generator polynomial of an Euclidean geometry code are specified.

The attractiveness of majority-logic decoding is its simple implementation. Several classes of majority-logic decodable block codes have been discovered for the past two decades. In this paper, a method of constructing a new class of majority-logic decodable block codes is presented. Each code in this class is formed by combining majority-logic decodable codes of shorter lengths. A procedure for orthogonalizing codes of this class is formulated. For each code, a lower bound on the number of correctable errors with majority-logic decoding is obtained. An upper bound on the number of orthogonalization steps for decoding each code is derived. Several majority-logic decodable codes that have more information digits than the Reed-Muller codes of the same length and the same minimum distance are found. Some results presented in this paper are extensions of the results of Lin and Weldon [11] and Gore [12] on the majority-logic decoding of direct product codes.

Cyclic product codes are useful for two reasons. First, they impart a great deal of algebraic structure to a subclass of the class of cyclic codes. Second, because they can be formulated in terms of much shorter (component) codes, their decoding may be considerably simpler than many other types of codes. In this paper both of the properties of cyclic product codes are developed. It is shown that the product of two majority-logic decodable cyclic codes is also majority-logic decodable provided that one of the component codes is one-step decodable. More precisely, if the row-component code can realize minimum distanced_1(i.e., correct[(d_1 -- 1)/2]errors) with a one-step majority-logic decoder and if the column-component code can realize minimum distanced_2with anL-step decoder, then the product code can realize distanced_1 d_2with anL-step decoder. It is also shown that the algebraic structure of cyclic product codes can be applied to establish the exact minimum distance of certain subclasses of BCH codes.

First it is shown that all binary Reed-Muller codes with one digit dropped can be made cyclic by rearranging the digits. Then a natural generalization to the nonbinary case is presented, which also includes the Reed-Muller codes and Reed-Solomon codes as special cases. The generator polynomial is characterized and the minimum weight is established. Finally, some results on weight distribution are given.

A class of cyclic codes is introduced by a polynomial approach that is an extension of the Mattson-Solomon method and of the Muller method. This class of codes contains several important classes of codes as subclasses, namely, BCH codes, Reed-Solomon codes, generalized primitive Reed-Muller codes, and finite geometry codes. Certain fundamental properties of this class of codes are derived. Some subclasses are shown to be majority-logic decodable.