Reduced-Complexity Guessing Codeword Decoding of BCH Codes with Most Reliable Cyclic Basis

In this paper, we consider a soft-input and soft-output iterative bounded-distance decoding (SISO-IBDD) algorithm in which the bounded-distance decodings are performed sequentially. And we study a computation method of a soft-output value to approximate Max-Log-MAP. In the decoding algorithm, search centers are generated by the sums of the hard-decision sequence obtained from a received sequence and test patterns which are selected beforehand, and the bounded-distance decodings with respect to the search centers are performed. We have proposed a selection method of test patterns for an iterative bounded-distance decoding (IBDD) algorithm in [5]. Simulation results have shown that the IBDD algorithm whose test patterns are selected by the proposed method with considerably small number of iterations provides better error performance than the Chase II decoding algorithm for several BCH codes over an AWGN channel using BPSK signaling. We adopt the selection method of test patterns for the SISO-IBDD algorithm and a computation method of a soft-output value which is modified based on the proposed method in [2] with respect to reliability factors. Over the AWGN channel using the BPSK signaling, we show bit error rates and average soft-output values given by the SISO-IBDD algorithm for the extended BCH (64, 36, 12) and (64, 45, 8) codes by computer simulation. For the extended BCH codes, the simulation results show that bit error rates of the SISO-IBDD algorithm are relatively smaller than those of a Chase-like decoding algorithm with the same number of iterations of the bounded-distance decodings and for each component bit, average soft-output values provided by the SISO-IBDD algorithm are close to those by the Max-Log-MAP decoding algorithm by computer simulation.

Irregular LDPC codes can achieve better error rate performance than regular LDPC codes. However, irregular LDPC codes have higher error floors than regular LDPC codes. The Ordered Statistic Decoding (OSD) algorithm achieves approximate Maximum Likelihood (ML) decoding. ML decoding is effective to lower error floors. However, the OSD estimates satisfy the parity check equation of the LDPC code even the estimates are wrong. Hybrid decoder combining LLR-BP decoding algorithm and the OSD algorithm cannot also lower error floors, because wrong estimates also satisfy the LDPC parity check equation. We proposed the concatenated code constructed with an inner irregular LDPC code and an outer Cyclic Redundancy Check (CRC). Owing to CRC, we can detect wrong codewords from OSD estimates. Our CRC-LDPC code with hybrid decoder can lower error floors in an AWGN channel. In wireless communications, we cannot neglect the effects of the channel. The OSD algorithm needs the ordering of each bit based on the reliability. The Channel State Information (CSI) is used for deciding reliability of each bit. In this paper, we evaluate the Block Error Rate (BLER) of the CRC-LDPC code with hybrid decoder in a fast fading channel with perfect and imperfect CSIs where 'imperfect CSI' means that the distribution of channel and those statistical average of the fading amplitudes are known at the receiver. By computer simulation, we show that the CRC-LDPC code with hybrid decoder can lower error floors than the conventional LDPC code with hybrid decoder in the fast fading channel with perfect and imperfect CSIs. We also show that combining error detection with the OSD algorithm is effective not only for lowering the error floor but also for reducing computational complexity of the OSD algorithm.

In this paper, we investigate the decoding of short Non-Binary (NB)-LDPC codes of rate 1/2 using a non-iterative approach based on the Maximum-Likelihood (ML) principle. The traditional decoding algorithms used to decode the NB-LDPC codes are by nature iterative where the Variables Nodes (VN) and Check Nodes (CN) exchange data iteratively during, at least, eight iterations which imposes a long decoding time to achieve good performance in terms of Frame Error Rate (FER). In this paper we propose a decoding algorithm based on the Maximum Likelihood (ML) search named Near ML approach where the number of tested words considered as potential codewords is highly reduced. Simulation of codes of lengths 16 and 48 are presented and the results show that the proposed algorithm achieves the performance offered by the EMS algorithm. The NB-LDPC of length 16 is shown to outperform the EMS algorithm.

Spatial modulation (SM) is a promising multiple- input multiple-output system used to increase spectral efficiency. The maximum likelihood (ML) decoder jointly detects the transmitted SM symbol, which is of high complexity. In this paper, a novel reliable sphere decoder (RSD) algorithm based on tree-search is proposed for the SM system. The basic idea of the proposed RSD algorithm is to reduce the size of the tree- search, and then, a smart searching method inside the reduced tree-search is performed to find the solution. The proposed RSD algorithm provides a significant reduction in decoding complexity compared to the ML decoder and existent decoders as well. Moreover, the RSD algorithm provides a flexible tradeoff between the bit error rate (BER) performance and decoding complexity, so as to be reliable for a wide range of practical hardware implementations. The BER performance and decoding complexity analysis for the RSD algorithm are studied, and Monte Carlo simulations are then provided to demonstrate the findings.

Space-time block codes from orthogonal designs proposed by Alamouti (1998), and Tarokh-Jafarkhani-Calderbank (199) have attracted much attention lately due to their fast maximum-likelihood (ML) decoding and full diversity. However, the maximum symbol transmission rate of a space-time block code from complex orthogonal designs for complex constellations is only 3/4 for three and four transmit antennas. Jafarkhani (see IEEE Trans. Commun., vol.49, no.1, p.1-4, 2001), and Tirkkonen-Boariu-Hottinen (see ISSSTA 2000, pp.429-432, September 2000) proposed space-time block codes from quasi-orthogonal designs, where the orthogonality is relaxed to provide higher symbol transmission rates. With the quasi-orthogonal structure, these codes still have a fast ML decoding, but do not have the full diversity. In this paper, we design quasi-orthogonal space-time block codes with full diversity by properly choosing the signal constellations. In particular, we propose that half symbols in a quasi-orthogonal design are from a signal constellation A and another half of them are optimal selections from the rotated constellation e/sup j/spl phi// A. The optimal rotation angles /spl phi/ are obtained for some commonly used signal constellations. The resulting codes have both full diversity and fast ML decoding.

In this paper, Sphere Decoding (SD) algorithms for Spatial Modulation (SM) are developed to reduce the computational complexity of Maximum-Likelihood (ML) detectors. Two SDs specifically designed for SM are proposed and analysed in terms of Bit Error Ratio (BER) and computational complexity. Using Monte Carlo simulations and mathematical analysis, it is shown that by carefully choosing the initial radius the proposed sphere decoder algorithms offer the same BER as ML detection, with a significant reduction in the computational complexity. A tight closed form expression for the BER performance of SM-SD is derived in the paper, along with an algorithm for choosing the initial radius which provides near to optimum performance. Also, it is shown that none of the proposed SDs are always superior to the others, but the best SD to use depends on the target spectral efficiency. The computational complexity trade-off offered by the proposed solutions is studied via analysis and simulation, and is shown to validate our findings. Finally, the performance of SM-SDs are compared to Spatial Multiplexing (SMX) applying ML decoder and applying SD. It is shown that for the same spectral efficiency, SM-SD offers up to 84% reduction in complexity compared to SMX-SD, with up to 1 dB better BER performance than SMX-ML decoder.

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.

In this paper, we propose a near maximum likelihood (ML) decoding technique, which reduces the computational complexity of the exact ML decoding algorithm. The computations needed for the tree search in the ML decoding is simplified by reducing the dimension of the search space prior to the tree search. In order to compensate performance loss due to the dimension reduction, a list stack algorithm (LSA) is considered, which produces a list of the top K closest points. The combination of both approaches, called reduced dimension list stack algorithm (RD-LSA), is shown to provide flexibility and offers a performance-complexity trade-off. Simulations performed for V-BLAST transmission demonstrate that significant complexity reduction can be achieved compared to the sphere decoding algorithm (SDA) while keeping the performance loss below an acceptable level.

The generalized minimum distance (GMD) and Chase (1972) decoding algorithms are some of the most important suboptimum bounded distance decoding algorithms for binary linear block codes over an additive white Gaussian noise (AWGN) channel. We compute the limitation of the ratio between the probability of decoding error for the GMD or any one of the Chase decoding algorithms and that of the maximum-likelihood (ML) decoding when the signal-to-noise ratio (SNR) approaches infinity. If the minimum Hamming distance of the code is greater than 2, the limitation is shown to be equal to 1 and thus the GMD and Chase decoding algorithms are asymptotically optimum.

Space-time block codes from orthogonal designs proposed by Alamouti, and Tarokh-Jafarkhani-Calderbank have attracted much attention lately due to their fast maximum-likelihood (ML) decoding and full diversity. However, the maximum symbol transmission rate of a space-time block code from complex orthogonal designs for complex constellations is only 3/4 for three and four transmit antennas. Recently, Jafarkhani,and Tirkkonen-Boariu-Hottinen proposed space-time block codes from quasi-orthogonal designs, where the orthogonality is relaxed to provide higher symbol transmission rates. With the quasi-orthogonal structure, these codes still have a fast ML decodng algorithm, but do not have the full diversity. The performance of these codes is better than that of the codes from orthogonal designs at low SNR, but worse at high SNR. This is due to the fact that the slope of the BER-SNR curve depends on the diversity. In this paper, we design quasi-orthogonal space-time block codes with full diversity by properly choosing the signal constellations. In particular, we propose that half of the symbols in a quasi-orthogonal design are from a signal constellation Α and another half of them are optimal selections from the rotated constellation ejφΑ . The optimal rotation angles φ are obtained for some commonly used signal contellations. The resulting codes have both full diversity and fast ML decoding. Simulation results show that the proposed codes outperform the codes from orthogonal designs at both low and high SNRs.© (2002) COPYRIGHT SPIE--The International Society for Optical Engineering. Downloading of the abstract is permitted for personal use only.

We study the maximum a posteriori (MAP) decoding of memoryless non-uniform sources over multiple-antenna channels. Our model is general enough to include space-time coding, BLAST architectures, and single-transmit multi-receive antenna systems which employ any type of channel coding. We derive a closed-form expression for the codeword pairwise error probability (PEP) of general multi-antenna codes using moment generating function and Laplace transform arguments. We then consider space-time orthogonal block (STOB) coding and prove that, similar to the maximum likelihood (ML) decoding case, detection of symbols is decoupled in MAP decoding. We also derive the symbol PEP in closed-form for STOB codes. We apply these results in several scenarios. First, we design a binary antipodal signaling scheme which minimizes the system bit error rate (BER) under STOB coding. At a BER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-6</sup> , this constellation has a channel signal-to-noise ratio (CSNR) gain of 4.7 dB over conventional BPSK signaling for a binary nonuniform source with p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> Delta <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">=</sub> P(0) = 0.9. We next design space-time linear dispersion (LD) codes which are optimized for the source distribution under the criterion of minimizing the union upper bound on the frame error rate (FER). Two codes are given here: one outperforms V-BLAST by 3.5 dB and Alamouti's code by 12.3 dB at an FER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-2</sup> for a binary source with p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> = 0.9, and the other outperforms V-BLAST by 4.2 dB at an FER of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> for a uniform source. These codes also outperform the LD codes of constructed under a different criteria. Finally, the problem of bit-to-signal mapping is studied. It is shown that for a binary source with p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> = 0.9, 64-QAM signaling, and SER = 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-3</sup> , a gain of 3.7 dB can be achieved using a better-than-Gray mapping. For a system with one transmit and two receive antennas that uses trellis coding with 16-QAM signaling, a 1.8 dB gain over quasi-Gray mapping and ML decoding is observed when MAP decoding is used for binary sources with p <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</sub> = 0.9.

Receiver complexity is an important performance metric in space time block coding (STBC). After comprehensive study and analysis of STBCs over complex domain, quaternion orthogonal designs (QODs) have been explored that can provide less decoder complexity. However, some researchers argue that existing maximum likelihood (ML) decoding rule for QODs does not yield optimal performance for quasi-orthogonal STBC-based QODs and, therefore, the design of optimal decoder for such designs remains a big challenge of this field. To counter this observation, we modify the system model slightly. In addition to this, we present a new generalized QOD construction that can generate QODs for any number of transmit antennas. These codes are quasi-orthogonal in complex domain and provide code rates higher than their respective complex orthogonal designs (CODs). We show that the proposed ML decoder provides optimal linear decoupled decoding solution for these codes because of their distinctive construction. Simulation results verify the optimal performance of the ML decoder for the proposed codes and also show that the diversity of the proposed quaternion-based codes is same as CODs, however, they provide better code rates.

The invention of Fountain codes is a major advance in the field of error correcting codes. The goal of this work is to study and develop algorithms for source and channel coding using a family of Fountain codes known as Raptor codes. From an asymptotic point of view, the best currently known sum-product decoding algorithm for non binary alphabets has a high complexity that limits its use in practice. For binary channels, sum-product decoding algorithms have been extensively studied and are known to perform well. In the first part of this work, we develop a decoding algorithm for binary codes on non-binary channels based on a combination of sum-product and maximum-likelihood decoding. We apply this algorithm to Raptor codes on both symmetric and non-symmetric channels. Our algorithm shows the best performance in terms of complexity and error rate per symbol for blocks of finite length for symmetric channels. Then, we examine the performance of Raptor codes under sum-product decoding when the transmission is taking place on piecewise stationary memoryless channels and on channels with memory corrupted by noise. We develop algorithms for joint estimation and detection while simultaneously employing expectation maximization to estimate the noise, and sum-product algorithm to correct errors. We also develop a hard decision algorithm for Raptor codes on piecewise stationary memoryless channels. Finally, we generalize our joint LT estimation-decoding algorithms for Markov-modulated channels. In the third part of this work, we develop compression algorithms using Raptor codes. More specifically we introduce a lossless text compression algorithm, obtaining in this way competitive results compared to the existing classical approaches. Moreover, we propose distributed source coding algorithms based on the paradigm proposed by Slepian and Wolf.

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.

Distributed orthogonal space-time block codes (DOSTBCs) achieving full-diversity order and single-symbol maximum-likelihood (ML) decodability have been introduced recently by Yi and Kim for cooperative networks, and an upper bound on the maximal rate of such codes along with code constructions has been presented. In this paper, a new class of single-symbol ML decodable precoded distributed space-time block codes (SSD-PDSTBCs) called semiorthogonal SSD-PDSTBCs (semi-SSD-PDSTBCs) is introduced wherein, the source performs linear precoding of information symbols appropriately before transmitting it to all the relays. It is shown that DOSTBCs are a special case of semi-SSD-PDSTBCs. A special class of semi-SSD-PDSTBCs having diagonal covariance matrix at the destination is studied and an upper bound on the maximal rate of such codes is derived. The bounds obtained are approximately twice larger than that of the DOSTBCs. A systematic construction of semi-SSD-PDSTBCs is presented when the number of relays <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> ges 4. The constructed codes are shown to achieve the upper bound on the rate when <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> is of the form <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">0</i> or <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</i> modulo <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</i> . For the rest of the values of <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K</i> , the constructed codes are shown to have rates higher than that of DOSTBCs. It is shown that semi-SSD-PDSTBCs cannot be constructed with any form of linear processing at the relays when the source does not perform linear precoding of the information symbols.