Extending the Dorsch decoder towards achieving maximum-likelihood decoding for linear codes

We consider an asynchronous cooperative communication system where two distributed transmitters are communicating with one destination. In this paper, a bounded delay-tolerant time interleave reversal Alamouti code (BDT-TIR AC) is proposed. BDT-TIR AC achieves full diversity, given that the path delay difference is within a tolerance limit. Furthermore, we design an efficient maximum-likelihood (ML) decoder. By employing a divide-and-conquer strategy, the original decoding problem is decoupled into several sub-problems in small size. In fact, these sub-problems contain either Alamouti code structure or only four symbols. By parallel decoding on the sub-problems, the proposed ML decoder exhibits low complexity advantage in large scale problems. Simulations of BDT-TIR AC confirm the full diversity gain. The bit error rate performance of BDT-TIR AC in the considered asynchronous scenarios is comparable with that of synchronized Alamouti code, and outperforms the latest delay-tolerant space-time code.

This letter considers the average complexity of maximum-likelihood (ML) decoding of convolutional codes. ML decoding can be modeled as finding the most probable path taken through a Markov graph. Integrated with the Viterbi algorithm (VA), complexity reduction methods often use the sum log likelihood (SLL) of a Markov path as a bound to disprove the optimality of other Markov path sets and to consequently avoid exhaustive path search. In this letter, it is shown that SLL-based optimality tests are inefficient if one fixes the coding memory and takes the codeword length to infinity. Alternatively, optimality of a source symbol at a given time index can be testified using bounds derived from log likelihoods of the neighboring symbols. It is demonstrated that such neighboring log likelihood (NLL)-based optimality tests, whose efficiency does not depend on the codeword length, can bring significant complexity reduction. The results are generalized to ML sequence detection in a class of discrete-time hidden Markov systems.

In this paper, we propose efficient maximum-likelihood (ML) decoding for binary Kronecker product-based (KPB) codes. This class of codes, have a matrix defined by the n-fold iterated Kronecker product Gn = F⊗n of a binary upper-triangular kernel matrix F, where some columns are suppressed given a specific puncturing pattern. Polar and Reed- Muller codes are well known examples of such KPB codes. The triangular structure of Gn enables to perform ML decoding as a binary tree search for the closest codeword to the received point. We take advantage of the highly regular fractal structure of Gn and the “tree folding” technique to design an efficient ML decoder, enabling to decode relatively longer block lengths than with a standard binary tree search. The tree κ-folding operation transforms the binary tree with N levels into a non-binary tree with N=2κ levels, where the search can be significantly accelerated by a suitable ordering of the branch metrics. For a given κ we can find (n over κ) different folding which lead to decoders with different complexity, for a given code. Using the proposed folded tree decoder, we provide exact ML performances of some Reed-Muller and polar codes over a binary AWGN channel for the block length up to 256.

The problem of decoding an (n, k, d) error-correcting block code, where a k-bit message word is mapped to an n-bit codeword, can be cast as a single-label classification problem. While it has been observed that the performance of such single-label neural decoders closely approaches that of the corresponding maximum likelihood decoder (MLD), the number of output nodes increases exponentially with k, making it prohibitive to implement for large k. To address this issue, we explore classification based multi-label neural decoders, in which the number of output nodes increases linearly with k. We consider well-known linear and non-linear block codes, as well as concatenated block codes, which have applications in emerging wireless networks. Our study finds that (i) although the number of output nodes linearly increases with k in a multi-label decoder, it requires more hidden layers and nodes in each hidden layer than the corresponding single-label decoder to achieve its best performance, and (ii) although one can design a multi-label decoder with bit error rate matching that of the MLD, it leaves more blocks in error leading to a reduced performance in terms of block error rate. We also note that the performance of the proposed decoder for concatenated codes is at least as good as that of a natural decoding algorithm in which the inner code is first decoded using the MLD and then the outer code is decoded with a polynomial-time decoding algorithm.

The error probability of maximum-likelihood (ML) soft-decision decoded binary block codes rarely accepts exact closed forms. In addition, for long codes ML decoding becomes prohibitively complex. Nevertheless, bounds on the performance of ML decoded systems provide insight into the effect of system parameters on the overall system performance in addition to a measure of efficiency of the sub-optimum decoding methods used in practice. In the article, a comprehensive study of a number of lower and upper bounds on the error probability of ML decoding of binary codes over AWGN channel is provided. Bounds considered here are bounds based on the so-called Bonferroni-type inequalities and bounds developed primarily in the light of the geometrical structure of the underlying signal constellations. The interrelationships among the bounds are explored and current tightest bounds at different noise levels are pointed out.

Maximum likelihood decoding of convolutional codes by Viterbi decoding method is an effective error correcting technique. However, little study on Viterbi decoding of high rate codes has been done since the number of paths entering each trellis state increases as the code rate increases. This paper presents a new maximum likelihood decoding of convolutional codes of rate (n ‐ 1)/n using a trellis. the trellis is drawn according to the state transitions of the syndrome former of the code, and has only one or two paths entering each state rather than 2n‐1 paths as in the case of conventional trellises. Thus, we can replace the 2n‐1‐ary comparisons at each state needed in the conventional Viterbi decoding for the code of rate (n ‐ 1)/n by binary comparisons. the implementation of maximum likelihood decoders for high rate convolutional codes is simplified greatly by this approach. Moreover, this decoding method shows the same performance as that of conventional Viterbi decoding method from the viewpoint of selecting the maximum likelihood code sequence.

In this paper, we introduce stopping sets for iterative row-column decoding of product codes using optimal constituent decoders. When transmitting over the binary erasure channel (BEC), iterative row-column decoding of product codes using optimal constituent decoders will either be successful, or stop in the unique maximum-size stopping set that is contained in the (initial) set of erased positions. Let C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> denote the product code of two binary linear codes C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> and C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> of minimum distances dc and d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> and second generalized Hamming weights d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ) and d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> ), respectively. We show that the size s <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> of the smallest noncode- word stopping set is at least mm(d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ),d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> )) > d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> , where the inequality follows from the Griesmer bound. If there are no codewords in C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</sub> with support set S, where S is a stopping set, then S is said to be a noncodeword stopping set. An immediate consequence is that the erasure probability after iterative row-column decoding using optimal constituent decoders of (finite-length) product codes on the BEC, approaches the erasure probability after maximum-likelihood decoding as the channel erasure probability decreases. We also give an explicit formula for the number of noncodeword stopping sets of size s <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">min</sub> , which depends only on the first nonzero coefficient of the constituent (row and column) first and second support weight enumerators, for the case when d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> ) < 2d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">r</sub> and d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> (C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ) < 2d <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> . Finally, as an example, we apply the derived results to the product of two (extended) Hamming codes and two Golay codes.

A systematic design of a trellis-based maximum-likelihood soft-decision decoder for linear block codes is presented. The essence of the decoder is to apply an efficient search algorithm for the error pattern on a reduced trellis representation of a certain coset. Rather than other efficient decoding algorithms, the proposed decoder is systematically designed for long codes, as well as for short codes. Computational gain of up to 6 is achieved for long high-rate codes over the well-known trellis decoder of Wolf (1978). Efficient decoders are also obtained for short and moderate length codes. >

In this paper, soft-decision decoding is considered as an optimisation problem and a computationally efficient soft-decision decoding algorithm for block codes is developed using a compact Genetic Algorithm. The performance of this algorithm is investigated and compared with other decoding schemes. The results show the proposed algorithm gives large gains over the Generalised Minimum Distance (GMD) decoding algorithm and algebraic hard-decision decoding in flat Rayleigh fading. Further, the proposed algorithm achieves near maximum likelihood decoder (MLD) performance for the codes simulated. The complexity of this algorithm is also discussed and compared to the standard Order-l reprocessing algorithm.

In this work, we revisit the structure of weight matrices for Linear Dispersion STBCs to admit ML decoding with low-complexity. We first propose novel sufficient design criteria for linear STBCs considering an arbitrary number of antennas and an arbitrary coding rate. Then we apply the derived criteria to three families of codes, multi-group decodable, fast decodable, and fast-group decodable codes. We provide analytical proofs showing that the ML-decoding complexity of such codes depends only on the weight matrices and their ordering and not on the channel gains or the number of antennas and explaining why the so far used Hurwitz-Radon theory-based approaches do not exactly determine the complexity of all classes of STBCs under ML decoding.

The problem of efficient maximum-likelihood soft decision decoding of binary BCH codes is considered. It is known that those primitive BCH codes whose designed distance is one less than a power of two, contain subcodes of high dimension which consist of a direct-sum of several identical codes. The authors show that the same kind of direct-sum structure exists in all the primitive BCH codes, as well as in the BCH codes of composite block length. They also introduce a related structure termed the concurring-sum, and then establish its existence in the primitive binary BCH codes. Both structures are employed to upper bound the number of states in the minimal trellis of BCH codes, and develop efficient algorithms for maximum-likelihood soft decision decoding of these codes. >

The block error probability under maximum likelihood decoding of general linear block codes and code ensembles over the q-ary symmetric channel is analyzed. The analysis makes use of two types of upper bounds and it is based on the codes (code ensembles) distance spectra. The first bound generalizes Poltyrev's union bound for the binary symmetric channel. The second bound relies on typical pairs decoding. The two bounds are developed in an asymptotic setting to analyze the behavior of linear block code ensembles, in terms of error exponent and maximum-likelihood decoding threshold. A sphere-packing lower bound is also developed, to provide a reference for the finite-length performance.

Bounds for the minimum distance and the error probability of decoding for tail biting convolutional codes are presented. We propose a decoding algorithm for these codes and obtain a bound for the asymptotical complexity of almost maximum likelihood decoding.

This paper presents the performance improvement of dual hop system formed by regenerative relay. The data encoded by orthogonal space time block codes (OSTBC), are sent through n transmit antennas simultaneously. The signal received at each receive antenna is sum of n transmitted signals added by the noise in their channel. Maximum likelihood decoding is used to decode the data. Dual hop is formed by placing a regenerative relay in between transmitter and receiver. The data from the first hop is decoded and forward to the second hop by properly encoding. The existing performance studies of the dual hop transmission and OSTBC transmissions individually. The OSTBC transmission through a regenerative relay is not well investigated so far. We evaluate the encoding and decoding of various codes and provided the simulation results for different orders of OSTBC with different modulations like QPSK, 16-QAM, 32-QAM and 64-QAM over a regenerative relay. Results provide the remarkable performance at the cost of about no extra processing.

An efficient decoder for the generalized first-order Reed-Muller code RM/sub q/ (1, m) is essential for the decoding of various block-coding schemes for orthogonal frequency-division multiplexing with reduced peak-to-mean power ratio. We present an efficient and simple maximum-likelihood decoding algorithm for RM/sup q/ (1, m). It is shown that this algorithm has lower complexity than other previously known maximum-likelihood decoders for RM/sub q/ (1, m).