Decoding Of Linear Block Codes Research Articles

New Golay decoding method using auto-encoder and OSD

In this study, we explore the potential of leveraging machine learning techniques, specifically auto-encoders (AE), for the decoding of linear block codes. Our findings suggest that this approach can outperform the conventional ordered statistics decoding (OSD) method, especially in a Rayleigh fading channel environment. We have rigorously trained the AE under both additive white Gaussian noise and Rayleigh fading channel conditions to ensure robustness in its performance. The output of the AE is combined with the received vector in a suitable manner to perform OSD. Through our experiments, we demonstrate that this proposed decoding approach yields better results than the conventional OSD method in Rayleigh fading channel when we used (23,12) Golay code.

Iterative Basis Update for Ordered Statistics Decoding of Linear Block Codes

Iterative Basis Update for Ordered Statistics Decoding of Linear Block Codes

Deep Learning-Based Decoding of Linear Block Codes for Spin-Torque Transfer Magnetic Random Access Memory

Thanks to its superior features of fast read/write speed and low power consumption, spin-torque transfer magnetic random access memory (STT-MRAM) has become a promising non-volatile memory (NVM) technology that is suitable for many applications. However, the reliability of STT-MRAM is seriously affected by the variation of the memory fabrication process and the working temperature, and the later will lead to an unknown offset of the channel. Hence, there is a pressing need to develop more effective error correction coding techniques to tackle these imperfections and improve the reliability of STT-MRAM. In this work, we propose, for the first time, the application of deep-learning (DL)-based algorithms and techniques to improve the decoding performance of linear block codes with short codeword lengths for STT-MRAM. We formulate the belief propagation (BP) decoding of linear block code as a neural network (NN) and propose a novel neural normalized-offset reliability-based min-sum (RB-MS) (NNORB-MS) decoding algorithm. We successfully apply our proposed decoding algorithm to the STT-MRAM channel through channel symmetrization to overcome the channel asymmetry. We also propose an NN-based soft information generation method (SIGM) to take into account the unknown offset of the channel. Simulation results demonstrate that our proposed NNORB-MS decoding algorithm can achieve significant performance gain over both the hard-decision decoding (HDD) and the regular RB-MS decoding algorithm, for cases without and with the unknown channel offset. Moreover, the decoder structure and time complexity of the NNORB-MS algorithm remain similar to those of the regular RB-MS algorithm.

Improved Decoding of Linear Block Codes by Concatenated Decoders

The use of decoding algorithms allows us to retrieve information after transmitting it over a noisy communication channel. Soft decision decoding is powerful in concatenation schemes that use two or more levels of decoding. In our case, we make a concatenation between the Hartmann & Rudolph (HR) algorithm as symbol-by-symbol decoder and the chase-2 algorithm that is word-to-word decoding algorithm. In this paper, we propose to combine two decoding algorithms for constructing a new one with more efficiency and less complexity. This work consists firstly to use the HR with a reduced number of codewords of the dual code then the Chase-2 algorithm which exploits the output of PHR. The simulations results and the comparisons made show that the proposed decoding scheme guarantees very good performance with reduced temporal complexity.

Fast and Scalable Soft Decision Decoding of Linear Block Codes

Ordered statistics-based decoding (OSD), which exhibits a near maximum likelihood decoding performance, suffers from huge computational complexity as the order increases. In this letter, we propose a fast and scalable OSD by considering the OSD as a fast searching problem. In the searching process, if the up-to-date minimum cost value is less than a predicted threshold value, then we can safely skip the search for remaining higher orders. The computational complexity of the proposed algorithm converges quickly to that of order one, regardless of the maximum order. Compared with the probabilistic necessary conditions-based OSD, the proposed algorithm exhibits speed-up gains of a factor of approximately 2,740 (at 3.0dB) for (127,64) BCH codes, with an indistinguishable decoding performance.

Особливості програмної реалізації біоінспірованого методу м’якого декодування лінійних блокових кодів

The important part of many telecommunication technologies are linear block codes. The use of soft decoding can increasing the effectiveness of these codes. The expediency of application bioinspired approach for soft decoding relatively long linear block codes is grounded. The general principles bioinspired approach to soft decoding of linear block codes is considered. The basic steps and flowchart of the bioinspired method of decoding linear block codes are presented. A distinctive feature of the proposed method of decoding linear block codes is the use of generalized bioinspired search to determine the most likely base information vector and corresponding bipolar codeword. It is noted that the choice of a particular procedure bioinspired search should be carried out according to the parameters of the code, channel characteristics and performance requirements decoding. The ways of software implementation bioinspired method of decoding linear block codes are determined. Bioinspired decoding algorithm for linear block codes and features its implementation are presented. The list of input and output data for this algorithm is presented. Principles of the subprocess of the most reliable basis generative matrix and subprocess bioinspired search basic information word (and the corresponding bipolar codeword) are presented.

Improved Decoding of linear Block Codes using compact Genetic Algorithms with larger tournament size

Improved Decoding of linear Block Codes using compact Genetic Algorithms with larger tournament size

Метод декодування лінійних блокових кодів на основі популяційних процедур пошукової оптимізації

It is shown that the classical methods of decoding linear block codes are based on the solution of algebraic equations and provide hard decisions. Thus, these decoding methods have low correction ability so do not satisfy the requirements of modern telecommunications systems. It is shown that the lack of non-algebraic decoding methods of linear block codes for ordered statistics is relatively high computational complexity, which makes it suitable only for short length codes. The necessity to improve the efficiency of decoding linear block codes used in modern telecommunications systems has been grounded. The variants of representing the problem of linear block codes decoding depending on the communication channel model have been considered. It is shown that this problem is a problem of discrete optimization, the dimension of which can be reduced by using the most reliable basis based on information about the reliability of the elements of received vector. The objective function which is advisable to used in the decoding of linear block codes is presented. Decoding method of linear block codes based on the joint use of the most reliable basis, objective function and population procedures of search optimization has been proposed

Efficient Decoding of Short Length Linear Cyclic Codes

Iterative soft decision decoding of linear block codes is a practical necessity when working with even modest block lengths. A number of algorithms have been proposed in the literature which use the permutation group of a code and the belief propagation (BP) algorithm for decoding. A novel soft-input, soft-output algorithm is presented that can be used for efficiently decoding of linear cyclic codes. Utilising the automorphism property of cyclic codes the permutation is incorporated into the belief propagation algorithm resulting in faster convergence and better error correcting performance. Performance of the new approach is analysed using a (63,45) BCH code and a (72,36) quadratic residue code. Simulation results show significant reduction in the average number of required decoding iterations and some improvement in error correcting performance over published algorithms.

Area-Efficient Parallel Syndrome Generators for Linear Block Codes

The common subexpression elimination (CSE) method has been applied to reduce the computational cost of the parallel syndrome generators, which are essential components for decoding of linear block codes. In this work, we present an alternative expression for the parallel syndrome generation formula, with which CSE is able to search redundant computations in a maximally expanded space to achieve a significantly higher optimization efficiency. In particular, two search space expansion schemes are studied for syndrome generators of BCH and RS codes. Experiments demonstrate at most 81 % of area savings using the proposed optimization schemes, and the post-layout results show the power saving of 41 ~ 45 % compared to a straightforward design.

Design of an Efficient Maximum Likelihood Soft Decoder for Systematic Short Block Codes

Maximum likelihood soft-decision decoding of linear block codes is addressed in this correspondence. A novel algorithm based on Chase-2 algorithm for the decoding of systematic binary block codes is detailed. A double re-encoding technique in place of the classical algebraic decoding for the computation of the candidate codeword list is the major innovation of the proposed algorithm. This approach has been successfully applied to systematic block codes that have a code rate equal to 1/2 and a parity check matrix composed of an invertible submatrix for the redundancy part. Simulation results show performance close to the optimum maximum likelihood decoding for an excellent tradeoff between BER performance and computational complexity. Then, the challenging issue of designing a decoder for a specific family of short binary block codes, called Cortex codes is also described. Three soft decoders for Cortex codes with lengths equal to 32, 64, and 128 and a code rate equal to 1/2 have been designed. Then, all the decoders were successively implemented onto an field-programmable gate array (FPGA) device. To our knowledge, they are the first efficient digital implementations of Cortex codes.

Majority Voting Procedure Allowing Soft Decision Decoding of Linear Block Codes on Binary Channels

In this paper we present an efficient algorithm to decode linear block codes on binary channels. The main idea consists in using a vote procedure in order to elaborate artificial reliabilities of the binary received word and to present the obtained real vector r as inputs of a SIHO decoder (Soft In/Hard Out). The goal of the latter is to try to find the closest codeword to r in terms of the Euclidean distance. A comparison of the proposed algorithm over the AWGN channel with the Majority logic decoder, Berlekamp-Massey, Bit Flipping, Hartman-Rudolf algorithms and others show that it is more efficient in terms of performance. The complexity of the proposed decoder depends on the weight of the error to decode, on the code structure and also on the used SIHO decoder.

A Note on the Branch-and-Cut Approach to Decoding Linear Block Codes

Maximum likelihood (ML) decoding of linear block codes can be considered as an integer linear programming (ILP). Since it is an NP-hard problem in general, there are many researches about the algorithms to approximately solve the problem. One of the most popular algorithms is linear programming (LP) decoding proposed by Feldman et al. LP decoding is based on the LP relaxation, which is a method to approximately solve the ILP corresponding to the ML decoding problem. Advanced algorithms for solving ILP (approximately or exactly) include cutting-plane method and branch-and-bound method. As applications of these methods, adaptive LP decoding and branch-and-bound decoding have been proposed by Taghavi et al. and Yang et al., respectively. Another method for solving ILP is the branch-and-cut method, which is a hybrid of cutting-plane and branch-and-bound methods. The branch-and-cut method is widely used to solve ILP, however, it is unobvious that the method works well for the ML decoding problem. In this paper, we show that the branch-and-cut method is certainly effective for the ML decoding problem. Furthermore the branch-and-cut method consists of some technical components and the performance of the algorithm depends on the selection of these components. It is important to consider how to select the technical components in the branch-and-cut method. We see the differences caused by the selection of those technical components and consider which scheme is most effective for the ML decoding problem through numerical simulations.

Performance of sphere decoding of block codes

A sphere decoder searches for the closest lattice point within a certain search radius. The search radius provides a tradeoff between performance and complexity. We focus on analyzing the performance of sphere decoding of linear block codes. We analyze the performance of soft-decision sphere decoding on AWGN channels and a variety of modulation schemes. A hard-decision sphere decoder is a bounded distance decoder with the corresponding decoding radius. We analyze the performance of hard-decision sphere decoding on binary and <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary symmetric channels. An upper bound on the performance of maximum-likelihood decoding of linear codes defined over <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F_q</i> (e.g. Reed- Solomon codes) and transmitted over <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> -ary symmetric channels is derived and used in the analysis. We then discuss sphere decoding of general block codes or lattices with arbitrary modulation schemes. The tradeoff between the performance and complexity of a sphere decoder is then discussed.

Stochastic Decoding of Linear Block Codes With High-Density Parity-Check Matrices

<para xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> This correspondence extends the application of the recently proposed stochastic decoding approach to decode linear block codes with high-density parity-check matrices and discusses its hardware complexity. Results demonstrate decoding performance close to floating-point iterative soft-input soft-output (SISO) decoding while offering nodes with considerably lower complexity compared to fixed-point SISO decoding. </para>

Selective trellis pruning for block codes: theory and application to iterative decoding

AbstractIn this paper we present a new method to simplify trellis‐based decoding of linear block codes. Our method is based on removing (pruning) some edges and states from the trellis representation of the code; the trellis is pruned through some hard‐decisions that are made on received bits whose likelihood exceeds a predefined threshold. We show in this paper how this simplification can be accounted for by a new generator matrix (and sometimes a coset leader) that totally parametrize the new trellis, or, equivalently, the set of allowed codewords after simplification. Extensive simulations show that significant computational savings can be achieved, at a very small loss in coding performance, as long as the operating point and threshold are carefully chosen. Moreover, the application of this technique to iterative decoding of product codes is outlined, and our results show that the simplifications do not hinder the convergence, again, as long as proper parameters are chosen. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Non‐binary SOVA algorithms for decoding of block codes on a sectionalized trellis

AbstractIn this paper, we explore computationally efficient implementations of the soft output viterbi algorithm (SOVA) applied to Soft‐Input Soft‐Output (SISO) decoding of linear block codes. In order to simplify the trellis‐based decoding of binary block codes with SOVA, we use the technique of sectionalization of the trellis, which has been successfully applied to the simplification of the MAP and Max‐Log‐MAP algorithms. Due to the branch complexity of the sectionalized trellis, we define a generalization of a non‐binary version of SOVA. However, the computational complexity of directly applying this approach remains too high for efficient implementation; we thus introduce the concept of non‐binary SOVA (NSOVA) with propagation of bit‐level reliabilities (BLR). This new algorithm is analyzed from a computational complexity viewpoint. Both serial and parallel implementations are explored. Finally, optimal sectionalizations are derived for selected codes; since the normal SOVA decoding is a particular case of NSOVA with BLR, we show that our approach is more efficient than a bit‐level trellis by showing that, for all the codes tested, the optimal trellis is a sectionalized one. Copyright © 2007 John Wiley &amp; Sons, Ltd.

Soft-Decision Decoding of Linear Block Codes Using Preprocessing and Diversification

Order-w reprocessing is a suboptimal soft-decision decoding approach for binary linear block codes in which up to w bits are systematically flipped on the so-called most reliable (information) basis (MRB). This correspondence first incorporates two preprocessing rules into order-w reprocessing and shows that, with appropriate choice of parameters, the proposed order-w reprocessing with preprocessing requires comparable complexity to order-w reprocessing but achieves asymptotically the performance of order-(w+2) reprocessing. To complement the MRB, a second basis is employed for practical SNRs and this approach is systematically extended to a multibasis order-w reprocessing scheme for high signal-to-noise ratios (SNRs). It is shown that the proposed multibasis scheme significantly enlarges the error-correction radius, a commonly used measure of performance at high SNRs, over the original (single-basis) order-w reprocessing. As a by-product, this approach also precisely characterizes the asymptotic performance of the well-known Chase and generalized minimum distance (GMD) decoding algorithms. The proposed algorithm successfully decodes the (192,96) Reed-Solomon concatenated code and the (256,147) extended BCH code in near optimal manner (within 0.01 dB at a block-error rate of 10 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-5</sup> ) with affordable computational cost

Reduced complexity maximum likelihood decoding of linear block codes

Reduced complexity maximum likelihood decoding of linear block codes

Evaluating viterbi decoded reed-solomon block codes on a complex spreaded DS/SSMA CDMA system

In recent years, the Viterbi Algorithm (VA) has been shown to be an efficient trellis decoder for binary and non-binary linear blocks. In this simulation study the performance of VA decoded non-binary Reed-Solomon (RS) block codes, employed in a wideband Code Division Multiple Access (CDMA) system, operating in multipath fading channel conditions, is considered. The RAKE receiver-based Direct Sequence Spread Spectrum Multiple Access (DS/SSMA) Quadrature Phase Shift Keying (QPSK) system employed in this study uses Complex Spreading Sequence (CSS), such as Zadoff-Chu (ZC), Double Sideband (DSB) Constant Envelope Linearly Interpolated Root-of-Unity (CE-LI-RU) filtered General Chirp-like (GCL), Analytical Bandlimited Complex (ABC) and Quadriphase (QPH) sequences. This paper firstly describes the notion of VA decoding of linear block codes. Background information is then given on the filtered and unfiltered CSS families employed in the simulation study. Next, the complex Direct Sequence Spread Spectrum Multiple Access (DS/SSMA) Quadrature Phase Shift Keying (QPSK) transmitter and RAKE receiver simulator structures, as well as a novel complex multipath fading channel simulator structure are presented. This is then followed by a description of this study's versatile multi-user multipath fading performance evaluation platform, which is constructed using the complex RAKE receiver-based DS/SSMA QPSK communication system and complex multipath fading channel simulator structures. Lastly, simulated Bit Error Rate (BER) performances are presented for uncoded wideband systems and wideband systems employing VA decoded non-binary RS block codes in realistic multi-user multipath fading channel conditions. The effect of Multi-User Interference (MUI), as well as that of Channel State Information (CSI) on the decoding performance of the VA are also considered.