Maximum-likelihood Decoding Performance Research Articles

Neural layered min-sum decoders for cyclic codes

This paper proposes a low-complexity neural network decoder based on the layered min-sum algorithm to decode cyclic codes. By generalizing the layered min-sum algorithm to its neural network counterpart, the number of network weights decreases while retaining a good error correction performance. The Bose–Chaudhuri–Hocquenghem (BCH) codes, quadratic residue (QR) codes, and punctured Reed–Muller (RM) codes are selected as three exemplary binary cyclic codes. Simulation results show that the proposed neural decoder achieves superior performance with less computational complexity compared with the state-of-the-art neural network decoder. Further, a neural decoder incorporating the modified random redundant decoding (mRRD) algorithm is investigated to approach the performance of maximum-likelihood decoding for some short codes.

High Speed Decoding for High-Rate and Short-Length Reed–Muller Code Using Auto-Decoder

In this paper, we show that applying a machine learning technique called auto-decoder (AD) to high-rate and short length Reed–Muller (RM) decoding enables it to achieve maximum likelihood decoding (MLD) performance and faster decoding speed than when fast Hadamard transform (FHT) is applied in additive white Gaussian noise (AWGN) channels. The decoding speed is approximately 1.8 times and 125 times faster than the FHT decoding for R(1,4) and R(2,4), respectively. The number of nodes in the hidden layer of AD is larger than that of the input layer, unlike the conventional auto-encoder (AE). Two ADs are combined in parallel and merged together, and then cascaded to one fully connected layer to improve the bit error rate (BER) performance of the code.

On Decoding of Reed-Muller Codes Using a Local Graph Search

We present a novel iterative decoding algorithm for Reed-Muller (RM) codes, which takes advantage of a graph representation of the code. Vertices of the considered graph correspond to codewords, with two vertices being connected by an edge if and only if the Hamming distance between the corresponding codewords equals the minimum distance of the code. The algorithm uses a greedy local search to find a node optimizing a metric, e.g. the correlation between the received vector and the corresponding codeword. In addition, the cyclic redundancy check can be used to terminate the search as soon as a valid codeword is found, leading to an improvement in the average computational complexity of the algorithm. Simulation results for both binary symmetric channel and additive white Gaussian noise channel show that the presented decoder approaches the performance of maximum likelihood decoding for RM codes of length less than 1024 and for the second-order RM codes of length less than 4096. Moreover, it is demonstrated that the considered decoding approach outperforms state-of-the-art decoding algorithms of RM codes with similar computational complexity for a wide range of block lengths and rates.

Rate‐compatible systematic polar codes

Puncturing and shortening are two common ways to achieve rate-compatible non-systematic polar codes (NSPCs). Systematic polar codes (SPCs) have been shown to outperform NSPCs with the same encoding and decoding complexity. However, rate-compatible SPCs have never been comprehensively studied in previous work. In this paper, two rate-compatible algorithms for SPCs are first proposed: uniform puncturing (UP) algorithm and uniform shortening (US) algorithm, which are referred to as SPC-UP and SPC-US, respectively. In order to effectively estimate the maximum likelihood decoding performance of punctured and shortened polar codes, subsequently, a distance spectrum calculation algorithm based on successive cancellation list (SCL) decoder for rate-compatible polar codes is proposed. Simulation results show that rate-compatible SPCs yield better bit error rate performance than rate-compatible NSPCs while they have the same frame error rate performance under different code rates and decoding algorithms. Eventually, union bounds that are obtained by the distance spectrum to provide the theoretical explanation for the superiority of rate-compatible SPCs are utilised.

Sphere Constraint Based Enumeration Methods to Analyze the Minimum Weight Distribution of Polar Codes

As an important performance metric, the minimum weight distribution (MWD) determines the maximum-likelihood decoding performance. In this paper, we focus on the MWDs of polar codes and concatenated polar codes to analyze the performance and optimize the encoding structure. Existing enumeration methods cannot guarantee to get the MWD exactly and have high search complexity. In order to enumerate all the codewords belonging to the MWD, the sphere constraint property is applied, which means that the codewords with the identical Hamming weight are distribute on a spherical shell. Based on this, three enumeration methods are proposed in this paper. We first propose a sphere constraint based enumeration method (SCEM) to analyze the MWD of polar codes with moderate complexity. Then, according to the SCEM and the Plotkin's construction of polar codes, a sphere constraint based recursive enumeration method (SCREM) is proposed to recursively calculate the MWD with a lower complexity. Finally, we propose a parity-check SCEM (PC-SCEM) to analyze the MWD of concatenated polar codes by introducing the parity-check equations of outer codes. Moreover, due to the sphere constraint property, the proposed three methods can exactly enumerate all the codewords with the minimum Hamming weight. The enumeration results show that the SCREM has about $10^6$ times complexity reduction compared with the existing methods at code length 128 and the PC-SCEM can be used to design high-performance CRC-polar concatenated codes.

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.

Theoretical Performance Evaluation of UHF-RFID Systems With Multi-Antenna Maximum-Likelihood Decoding

Radio frequency identification (RFID) is a technique that enables autonomous identification of objects using tiny and cheap transponders (tags). Therefore, its use increases, especially in the area of logistics. However, in dense applications scenarios with many tags multiple tags may transmit their data at the same time, leading to collisions. These collisions, especially in case of UHF RFID, significantly reduce the system throughput. The use of multiple input–multiple output (MIMO) is one candidate to achieve a high throughput even in case of tag collisions. Within this paper, we therefore analyze the performance of maximum-likelihood decoding for UHF RFID with multiple reception antennas. We especially take into account the impact of the FM0 channel encoding used by the EPCglobal Class-1 Gen-2 standard, the most common UHF RFID standard in the area of logistics. The channel characteristics of passive RFID systems follow the so-called double or backscatter Rayleigh distribution, which makes the performance analysis a challenging task. We use two methodologies to analyze the performance in this. First, we derive an upper bound symbol error probability (SEP) based on the pairwise error probability (PEP). The second technique depends on the exact evaluation of the SEP by taking into account all probabilities of errors between symbols. We finally compare the accuracy of both methodologies with simulation results. These results show that the PEP analysis converges with increasing number of antennas, and the exact analysis gives accurate results even with only two receive antennas.

A Cross-Layer Approach to Reducing Packet Delay in Polling-Based Multiuser Systems

We consider a system where users are polled for transmission by a central server, and the link layer of each user employs automatic repeat request (ARQ) to ensure error-free delivery of packets. We present a cross-layer approach for reducing average packet delay (queuing delay plus service time). It is known that forward error correction (FEC) schemes can potentially lower the average packet delay by reducing the packet error probability (PEP) and, hence, the number of retransmissions. However, more precise results in the literature are few and far between. In this paper, we first establish that relative to an uncoded system, it is sufficient to reduce the average service time (AST) using FEC to achieve lower average packet delay. We then study and quantify the reduction in AST that can be achieved using the best possible FEC codes. The specific findings and contributions from our work are as follows: 1) We provide several bounds on the reduction in AST using the best possible FEC codes; 2) we give a sufficient condition when no FEC scheme can reduce the AST; 3) for Gaussian channels, we find that a relatively high PEP $(\sim\!\!\text{10}^{-2})$ , which is obtained using as high a coding rate as possible, typically results in sufficiently small AST; 4) the performance of optimum maximum-likelihood decoding can be approached by a lower complexity bounded distance decoder; and 5) average packet delay can be further reduced in certain cases by opportunistically combining and encoding several packets jointly.

Interference Management via Sliding-Window Coded Modulation for 5G Cellular Networks

SWCM aims to mitigate the adverse effects of interference at the physical layer by tracking the optimal maximum likelihood decoding performance with low-complexity decoding and minimal coordination overhead. This article reviews the basic structure of the SWCM scheme built on the principles of network information theory, and discusses how it can be extended and implemented in practical wireless communication systems. Using a representative implementation based on LTE OFDM MIMO systems, extensive link-level and system-level performance simulations are carried out, which demonstrate that SWCM offers significant gains for all users over conventional interference-aware communication schemes. Network operating prerequisites for SWCM are also discussed to facilitate the standardization effort for its adoption in the fifth generation cellular network.

Distance Enumerator Analysis for Interleave-Division Multi-User Codes

Consider an interleave-division multi-user coding model for a Gaussian multiple-access channel (MAC). Each user’s message is encoded by a separate channel encoder followed by a user-interleaver, which is employed for user separation. All the users’ codewords are jointly regarded as one multi-user codeword, and the distance between two multi-user codewords is defined as the Euclidean distance after the inter-user superposition caused by the MAC. A distance enumerator analysis estimates the multi-user maximum likelihood decoding performance. By introducing user-scramblers, a multi-user code ensemble is defined, and its distance enumerator is calculated as concatenated multiple single-user codes and a superposition code. Both finite-length and asymptotic analyses are given and it is shown that interleave-division multi-user codes have much better distance properties than conventional random direct-sequence code-division multiple-access. In fact, the codes achieve almost the same asymptotic uniquely decodable performance as a multi-user random code.

Performance Analysis of Raptor Codes Under Maximum Likelihood Decoding

In this paper, we analyze the maximum likelihood decoding performance of Raptor codes with a systematic low-density generator-matrix code as the pre-code. By investigating the rank of the product of two random coefficient matrices, we derive upper and lower bounds on the decoding failure probability. The accuracy of our analysis is validated through simulations. Results of extensive Monte Carlo simulations demonstrate that for Raptor codes with different degree distributions and pre-codes, the bounds obtained in this paper are of high accuracy. The derived bounds can be used to design near-optimum Raptor codes with short and moderate lengths.

Low-Complexity Sphere Decoding of Polar Codes Based on Optimum Path Metric

Sphere decoding (SD) of polar codes is an efficient method to achieve the error performance of maximum likelihood (ML) decoding. But the complexity of the conventional sphere decoder is still high, where the candidates in a target sphere are enumerated and the radius is decreased gradually until no available candidate is in the sphere. In order to reduce the complexity of SD, a stack SD (SSD) algorithm with an efficient enumeration is proposed in this paper. Based on a novel path metric, SSD can effectively narrow the search range when enumerating the candidates within a sphere. The proposed metric follows an exact ML rule and takes the full usage of the whole received sequence. Furthermore, another very simple metric is provided as an approximation of the ML metric in the high signal-to-noise ratio regime. For short polar codes, simulation results over the additive white Gaussian noise channels show that the complexity of SSD based on the proposed metrics is up to 100 times lower than that of the conventional SD.

Bounds on Maximum Likelihood Decoding Performance for Linear Codes at Low Rates

For a given linear code (ensemble), upper bounds on the error probability under maximum likelihood decoding are investigated at low rates. A class of symmetric memoryless channels suitable for Bhattacharyya-type bounds, which are particularly of importance at low rates, is introduced. Over a symmetric channel, a lower bound on the error exponent is derived for a given linear code. A sufficient condition for achieving the expurgated exponent, which is the best among known error exponents at low rates, within a fixed discrepancy is given. Over a general discrete memoryless channel, the same analysis provides a lower bound on the average error exponent for an ensemble of coset codes generated by a given linear code. The bounding technique is extended to the case of generalized maximum likelihood decoding with erasure and list- decoding options.

New Techniques for Upper-Bounding the ML Decoding Performance of Binary Linear Codes

In this paper, new techniques are presented to either simplify or improve most existing upper bounds on the maximum-likelihood (ML) decoding performance of the binary linear codes over additive white Gaussian noise (AWGN) channels. Firstly, the recently proposed union bound using truncated weight spectrum by Ma et al. is re-derived in a detailed way based on Gallager's first bounding technique (GFBT), where the "good region" is specified by a sub-optimal list decoding algorithm. The error probability caused by the bad region can be upper-bounded by the tail-probability of a binomial distribution, while the error probability caused by the good region can be upper-bounded by most existing techniques. Secondly, we propose two techniques to tighten the union bound on the error probability caused by the good region. The first technique is based on pair-wise error probabilities. The second technique is based on triplet-wise error probabilities, which can be upper-bounded by the fact that any three bipolar vectors form a non-obtuse triangle. The proposed bounds improve the conventional union bounds but have a similar complexity since they involve only the Q-function. The proposed bounds can also be adapted to bit-error probabilities.

On the Probabilistic Computation Method with Reliability for the Weight Distribution of LDPC Codes

In the analysis of maximum-likelihood decoding performance of low-density parity-check (LDPC) codes, the weight distribution is an important factor. We presented a probabilistic method for computing the weight distribution of LDPC codes, and showed results of computing the weight distribution of several LDPC codes. In this paper, we improve our previously presented method and propose a probabilistic computation method with reliability for the weight distribution of LDPC codes. Using the proposed method, we can determine the weight distribution with small failure probability.

A low-complexity generalized sphere decoding approach for underdetermined linear communication systems: performance and complexity evaluation

For underdetermined linear systems, original sphere decoding (SD) algorithms fail due to zero diagonal elements in the upper-triangular matrix of the QR or Cholesky factorization of the underdetermined matrix. To solve this problem, this paper presents a low-complexity generalized sphere decoding (GSD) approach by transforming the original underdetermined problem into the full-column-rank one so that standard SD can be directly applied on the transformed problem. Since the introduced transformation maintains the dimension of the original problem for all M-QAM's, the proposed GSD approach provides significant reduction in complexity as compared to other GSD schemes, especially for M-QAM with large signaling constellation. Both performance and expected complexity are analyzed to provide the comprehensive relationships between the performance and complexity of the proposed GSD and its parameters. Illustrative simulation and analytical results are in good agreement in terms of both the performance and complexity and indicate that with the properly selected design parameters, the proposed GSD scheme can approach the optimum maximumlikelihood decoding (MLD) performance with low complexity for underdetermined linear communication systems including underdetermined MIMO systems, and the proposed expected complexity analysis can be used as reliable complexity estimation for practical implementation of the proposed algorithm and serve as reference for other GSD algorithms.

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.

Sparsely spread CDMA—a statistical mechanics-based analysis

Sparse code division multiple access (CDMA), a variation on the standard CDMA method in which the spreading (signature) matrix contains only a relatively small number of nonzero elements, is presented and analysed using methods of statistical physics. The analysis provides results on the performance of maximum likelihood decoding for sparse spreading codes in the large system limit. We present results for both cases of regular and irregular spreading matrices for the binary additive white Gaussian noise channel (BIAWGN) with a comparison to the canonical (dense) random spreading code.

Pseudo-Codeword Analysis of Tanner Graphs From Projective and Euclidean Planes

We consider coded data transmission over a binary-input output-symmetric memoryless channel using a binary linear code. In order to understand the performance of maximum-likelihood (ML) decoding, one studies the codewords, in particular the minimal codewords, and their Hamming weights. In the context of linear programming (LP) decoding, one's attention needs to be shifted to the pseudo-codewords, in particular, to the minimal pseudo-codewords and their pseudo-weights. In this paper, we investigate some families of codes that have good properties under LP decoding, namely certain families of low-density parity-check (LDPC) codes that are derived from projective and Euclidean planes: we study the structure of their minimal pseudo-codewords and give lower bounds on their pseudo-weight. Besides this main focus, we also present some results that hold for pseudo-codewords and minimal pseudo-codewords of any Tanner graph, and we highlight how the importance of minimal pseudo-codewords under LP decoding varies depending on which binary-input output-symmetric memoryless channel is used.

Reliability-Based Soft-Decision Decoding With Multiple Biases

In this paper, a new reliability-based soft-decision decoding algorithm is presented. This algorithm repeatedly uses biased reliability values to construct the most-reliable-basis (MRB). As a result, this new method makes use of multiple information sets in a stochastic way. Compared to previously proposed competitive approaches, this new method produces a more efficient MRB reprocessing type algorithm to achieve near maximum-likelihood decoding (MLD) performance with a proper choice of the bias value. It can be combined with any MRB reprocessing type algorithm and in each case, a tight performance analysis can be derived