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We propose a novel approach to optimization of irregular nonbinary (NB) quasi-cyclic (QC)-LDPC codes over small alphabets. In this approach, first, the base parity-check matrices are constructed by a simulated annealing method, and then these matrices are labeled by the field elements while maximizing the so- called generalized girth of the Tanner graph. In order to analyze the performance of the constructed irregular NB LDPC codes, a new ensemble of irregular NB LDPC codes over the extensions of the binary Galois field is introduced. A finite-length random coding bound on the error probability of the maximum-likelihood (ML) decoding over the binary phase shift keying (BPSK) input AWGN channel for the new code ensemble is derived. The frame error rate (FER) performance of the sum-product belief-propagation (BP) decoding of the constructed NB QC-LDPC block codes is compared to that of both the optimized binary QC-LDPC block codes in the 5G standard and the best known NB QC-LDPC codes as well as to the derived random coding bound on the ML decoding error probability. It is shown that the obtained bound predicts the behavior of BP decoding performance of practical NB QC-LDPC codes more accurately than the BP decoding thresholds do. [ABSTRACT FROM AUTHOR]

Stopping sets play a crucial role in failure events of iterative decoders over a binary erasure channel (BEC). The $\ell $ th stopping redundancy is the minimum number of rows in the parity-check matrix of a code, which contains no stopping sets of size up to $\ell $. In this paper, a notion of coverable stopping sets is defined. In order to achieve maximum-likelihood performance under iterative decoding over the BEC, the parity-check matrix should contain no coverable stopping sets of size $\ell $ , for $1 \le \ell \le n-k$ , where $n$ is the code length, $k$ is the code dimension. By estimating the number of coverable stopping sets, we obtain upper bounds on the $\ell $ th stopping redundancy, $1 \le \ell \le n-k$. The bounds are derived for both specific codes and code ensembles. In the range $1 \le \ell \le d-1$ , for specific codes, the new bounds improve on the results in the literature. Numerical calculations are also presented. [ABSTRACT FROM AUTHOR]

The performance of maximum-likelihood (ML) decoding on the binary erasure channel for finite-length low-density parity-check (LDPC) codes from two random ensembles is studied. A tightened union-type upper bound on the ML decoding error probability based on the precise coefficients of the average weight spectrum is presented. For LDPC codes from the Gallager ensemble and the Richardson–Urbanke ensemble, new upper bounds on the ML decoding performance based on computing the rank of submatrices of the code parity-check matrix are derived. A new lower bound on the ML decoding threshold followed from the latter error probability bound is obtained. An improved lower bound on the error probability for codes with a known estimate on the minimum distance is presented as well. A new low-complexity near-ML decoding algorithm for quasi-cyclic LDPC codes is proposed and simulated. Its performance is compared to the simulated belief propagation and ML decoding performance and simulated performance of the best known improved iterative decoding techniques, as well as, with the derived upper bounds on the ML decoding performance and with decoding thresholds obtained by the density evolution technique. [ABSTRACT FROM AUTHOR]

A new method is presented for low-complexity near-maximum-likelihood (ML) decoding of low-density parity-check (LDPC) codes over the additive white Gaussian noise channel. The proposed method termed belief-propagation–list erasure decoding (BP-LED) is based on erasing carefully chosen unreliable bits performed in case of BP decoding failure. A strategy of introducing erasures into the received vector and a new erasure decoding algorithm are proposed. The new erasure decoding algorithm, called list erasure decoding, combines ML decoding over the BEC with list decoding applied if the ML decoder fails to find a unique solution. The asymptotic exponent of the average list size for random regular LDPC codes from the Gallager ensemble is analyzed. Furthermore, a few examples of irregular quasi-cyclic LDPC as well as randomly constructed regular LDPC codes of short and moderate lengths are studied by simulations and their performance is compared to the tightened upper bound on the LDPC ensemble-average performance and the upper bound on the average performance of random linear codes under ML decoding. A comparison of the BP decoding and BP-LED performance of the WiMAX standard codes and performance of the near-ML BEAST decoding are presented. The new algorithm is applied to decoding a short nonbinary (NB) LDPC code over extensions of the binary Galois field. The obtained simulation results are compared to the tightened upper bound on the ensemble-average performance of the binary image of regular NB LDPC codes. [ABSTRACT FROM AUTHOR]

This paper studies an almost-lossless source-channel coding scheme in which source messages are assigned to different classes and encoded with a channel code that depends on the class index. The code performance is analyzed by means of random-coding error exponents and validated by simulation of a low-complexity implementation using existing source and channel codes. While each class code can be seen as a concatenation of a source code and a channel code, the overall performance improves on that of separate source-channel coding and approaches that of joint source-channel coding when the number of classes increases. [ABSTRACT FROM PUBLISHER]

A unified approach to search for and optimize codes determined by their sparse parity-check matrices is presented. Replacing the nonzero elements of a binary parity-check matrix (the base or parent matrix) either by circulants or by companion matrices of elements from a finite field GF (2^m) , we obtain quasi-cyclic low-density parity-check (LDPC) block codes and binary images of nonbinary LDPC block codes, respectively. By substituting monomials of a formal variable $D$ , we obtain the polynomial description of an LDPC convolutional code. A set of performance measures applicable to different classes of LDPC codes is considered, and a greedy algorithm for code performance optimization is presented. The heart of the new optimization algorithm is a fast procedure for searching for LDPC codes with large girth of their Tanner graphs. For a few classes of LDPC codes, examples of codes combining good error-correcting performance with compact representation are obtained. In particular, we present optimized convolutional LDPC codes and conclude that the LDPC block codes are still superior to their convolutional counterparts if both decoding complexity and coding delay are considered. Moreover, a specific channel model can easily be embedded into the optimization loop. Thereby, the code can be optimized for a specific channel. The efficiency of such an optimization is demonstrated via an example of faster than Nyquist (FTN) signaling using LDPC codes. The FTN strategy combined with a rate $R=1/2$ LDPC code of length 64 800 optimized for effective data rate $R=3/4$ gains more than 0.5 dB compared with the standard LDPC codes of the same rate and length. The obtained gain corresponds to transmission at the capacity of the binary input additive white Gaussian noise channel. In most numerical examples, we consider codes with bidiagonal structure of the parity-check matrix. This restriction preserves low encoding complexity and allows fair comparison with codes selected for communication standards. [ABSTRACT FROM PUBLISHER]

In 1995, Best published a formula for the exact bit error probability for Viterbi decoding of the rate R=1/2, memory m=1 (two-state) convolutional encoder with generator matrix G(D)=(1 \quad 1+D) when used to communicate over the binary symmetric channel. Their formula was later extended to the rate R=1/2, memory m=2 (four-state) convolutional encoder with generator matrix G(D)=(1+D^2 \quad 1+D+D^2) by Lentmaier In this paper, a different approach to derive the exact bit error probability is described. A general recurrent matrix equation, connecting the average information weight at the current and previous states of a trellis section of the Viterbi decoder, is derived and solved. The general solution of this matrix equation yields a closed-form expression for the exact bit error probability. As special cases, the expressions obtained by Best for the two-state encoder and by Lentmaier for a four-state encoder are used. The closed-form expression derived in this paper is evaluated for various realizations of encoders, including rate R=1/2 and R=2/3 encoders, of as many as 16 states. Moreover, it is shown that it is straightforward to extend the approach to communication over the quantized additive white Gaussian noise channel. [ABSTRACT FROM PUBLISHER]

The relation between parity-check matrices of quasi-cyclic (QC) low-density parity-check (LDPC) codes and biadjacency matrices of bipartite graphs supports searching for powerful LDPC block codes. Using the principle of tailbiting, compact representations of bipartite graphs based on convolutional codes can be found. Bounds on the girth and the minimum distance of LDPC block codes constructed in such a way are discussed. Algorithms for searching iteratively for LDPC block codes with large girth and for determining their minimum distance are presented. Constructions based on all-one matrices, Steiner Triple Systems, and QC block codes are introduced. Finally, new QC regular LDPC block codes with girth up to 24 are given. [ABSTRACT FROM AUTHOR]

A rate R = 5/20 hypergraph-based woven convolutional code with overall constraint length 67 and constituent convolutional codes is presented. It is based on a 3-partite, 3-uniform, 4-regular hypergraph and contains rate Rc = 3/4 constituent convolutional codes with overall constraint length 5. Although the code construction is based on low-complexity codes, the free distance of this construction, computed with the BEAST algorithm, is dfree = 120, which is remarkably large. [ABSTRACT FROM AUTHOR]

Constructions of woven graph codes based on constituent block and convolutional codes are studied. It is shown that within the random ensembles of such codes based on A-partite, s-uniform hypergraphs, where s depends only on the code rate, there exist codes satisfying the Gilbert--Varshamov (GV) and the Costello lower bound on the minimum distance and the free distance, respectively. A connection between regular bipartite graphs and tailbiting (TB) codes is shown. Some examples of woven graph codes are presented! Among them, an example of a rate Rwg = 1/3 woven graph code with dfree = 32 based on Heawood's bipartite graph, containing n = 7 constituent rate Rc = 2/3 convolutional codes with overall constraint lengths vc = 5, is given. [ABSTRACT FROM AUTHOR]

A Bidirectional Efficient Algorithm for Searching code Trees (BEAST) is proposed for efficient soft-output decoding of block codes and concatenated block codes. BEAST operates on trees corresponding to the minimal trellis of a block code and finds a list of the most probable codewords. The complexity of the BEAST search is significantly lower than the complexity of trellis-based algorithms, such as the Viterbi algorithm and its list generalizations. The outputs of BEAST, a list of best codewords and their metrics, are used to obtain approximate a posteriori probabilities (APPs) of the transmitted symbols, yielding a soft-input soft-output (SISO) symbol decoder referred to as the BEAST-APP decoder. This decoder is employed as a component decoder in iterative schemes for decoding of product and incomplete product codes. Its performance and convergence behavior are investigated using extrinsic information transfer (EXIT) charts and compared to existing decoding schemes. It is shown that the BEAST-APP decoder achieves performances close to the Bahl--Cocke--Jelinek-Raviv (BCJR) decoder with a substantially lower computational complexity. [ABSTRACT FROM AUTHOR]

List decoding of binary block codes for the additive white Gaussian noise (AWGN) channel is considered. The output of a list decoder is a list of the L most likely codewords, that is, the L signal points closest to the received signal in the Euclidean-metric sense. A decoding error occurs when the transmitted codeword is not on this list. It is shown that the list error probability is fully described by the so-called list configuration matrix, which is the Gram matrix obtained from the signal vectors forming the list. The worst case list configuration matrix determines the minimum list distance of the code, which is a generalization of the minimum distance to the case of list decoding. Some properties of the list configuration matrix are studied and their connections to the list distance are established. These results are further exploited to obtain a new upper bound on the list error probability, which is tighter than the previously known bounds. This bound is derived by combining the techniques for obtaining the tangential union bound with an improved bound on the error probability for a given list. The results are illustrated by examples. [ABSTRACT FROM AUTHOR]

An extended table of Shuurman's bounds on the state complexity of short binary linear codes is presented. Some new lower and upper bounds are obtained. Most of the newly found codes are based on the so-called double zero-tail termination (DZT) construction. [ABSTRACT FROM AUTHOR]

The slope of the active distances is an important parameter when investigating the error-correcting capability of convolutional codes and the distance behavior of concatenated convolutional codes. The slope of the active distances is equal to the minimum average weight cycle in the state-transition diagram of the encoder. A general upper bound on the slope depending on the free distance of the convolutional code and new upper bounds on the slope of special classes of binary convolutional codes are derived. Moreover, a search technique, resulting in new tables of rate R = 1/2 and rate R = 1/3 convolutional encoders with high memories and large active distance-slopes is presented. Furthermore, we show that convolutional codes with large slopes can be used to obtain new tailbiting block codes with large minimum distances. Tables of rate R = 1/2 and rate R = 1/3 tailbiting codes with larger minimum distances than the best previously known quasi-cyclic codes are given. Two new tailbiting codes also have larger minimum distances than the best previously known binary linear block codes with same size and length. One of them is also superior in terms of minimum distance to any previously known binary nonlinear block code with the same set of parameters. [ABSTRACT FROM AUTHOR]

Presents rational rate punctured convolutional codes. Search technique for the rational rate punctured convolutional codes; Properties and examples of rational rate punctured convolutional codes; Conclusions.