Maximum-likelihood Decoding Algorithm Research Articles

Low-depth random Clifford circuits for quantum coding against Pauli noise using a tensor-network decoder

Recent work [M. J. Gullans , ] has shown that quantum error correcting codes defined by random Clifford encoding circuits can achieve a nonzero encoding rate in correcting errors even if the random circuits on n qubits, embedded in one spatial dimension (1D), have a logarithmic depth d=O(logn). However, this was demonstrated only for a simple erasure noise model. In this work, we discover that, for the same class of codes, this desired property indeed holds for the conventional Pauli noise model. Specifically, we numerically demonstrate that the hashing bound, i.e., a rate known to be achieved with d=O(n)-depth random encoding circuits, can be attained for the above codes even when the circuit depth is restricted to d=O(logn) in 1D for depolarizing noise of various strengths. This analysis is made possible with our development of a tensor-network maximum-likelihood decoding algorithm that works efficiently for log-depth encoding circuits in 1D. Published by the American Physical Society 2024

On Ordinary Words of Standard Reed–Solomon Codes over Finite Fields

Reed–Solomon codes are widely used to establish a reliable channel to transmit information in digital communication which has a strong error correction capability and a variety of efficient decoding algorithm. Usually we use the maximum likelihood decoding (MLD) algorithm in the decoding process of Reed–Solomon codes. MLD algorithm relies on determining the error distance of received word. Dür, Guruswami, Wan, Li, Hong, Wu, Yue and Zhu et al. got some results on the error distance. For the Reed–Solomon code [Formula: see text], the received word [Formula: see text] is called an ordinary word of [Formula: see text] if the error distance [Formula: see text] with [Formula: see text] being the Lagrange interpolation polynomial of [Formula: see text]. We introduce a new method of studying the ordinary words. In fact, we make use of the result obtained by Y.C. Xu and S.F. Hong on the decomposition of certain polynomials over the finite field to determine all the ordinary words of the standard Reed–Solomon codes over the finite field of [Formula: see text] elements. This completely answers an open problem raised by Li and Wan in [On the subset sum problem over finite fields, Finite Fields Appl. 14 (2008) 911–929].

Energy-Efficient and Low-Latency Massive SIMO Using Noncoherent ML Detection for Industrial IoT Communications

To enable ultrareliable low-latency wireless communications required in the Industrial Internet of Things, in this paper we develop an energy-based modulation [i.e., non-negative pulse amplitude modulation (PAM)] constellation design framework for noncoherent detection in massive single-input multiple-output (SIMO) systems. We consider that one single-antenna transmitter communicates to a receiver with a large number of antennas over a Rayleigh fading channel, and the receiver decodes the transmitted information at the end of every symbol. For such an SIMO system with non-negative PAM modulation, we first propose a fast noncoherent maximum-likelihood decoding algorithm and derive a closed-form expression of its symbol error probability (SEP). We then enhance the system energy efficiency by finding the optimal PAM constellation that minimizes the exact SEP subject to a total signal power constraint for such a system with an arbitrary number of receiver antennas, signal-to-noise ratio (SNR), and constellation size. Furthermore, the closed-form upper and lower bounds on the optimal SEP are derived. Based on these bounds, the exact expression for coding gain of the dominant term of the SEP is presented for such an optimal massive SIMO system. We also present an asymptotic SEP expression at a high SNR regime and the approximate diversity gain of the system. Simulation results for the proposed optimal PAM constellation validate the theoretical analysis, and show that our presented optimal constellation attains significant performance gains over the currently available minimum-distance-based constellation systems.

Maximum likelihood decoding based on pseudo-captured image templates for image sensor communication

This paper focuses on an image sensor communication system that uses an LED as the transmitter and a high-speed image sensor (camera) as the receiver. Communication in this scheme depends on the quality of images transmitted from the LED to the sensor. If the image becomes unfocused on the way to the receiver, the LED luminance that make up the signal cannot be detected, so the receiver cannot demodulate the signal data. To overcome this problem, this study proposes a novel demodulation scheme to recover data from a degraded image, based on a maximum likelihood decoding (MLD) algorithm. The proposed method creates template images that imitate all possible blinking patterns produced by the LED transmitter, and then calculates the Euclidean distances between pixels in the captured image and the pseudo images for all possible blinking patterns. Finally, the algorithm chooses the image template with the smallest Euclidian distance from the received signal as the recovered data. Though an exhaustive set of image templates must be prepared for the proposed MLD, the number of templates depends on the number of LEDs on the transmitter. Thus, the computational complexity of this method increases as the number of transmitter LEDs increases. To reduce the computational complexity of the proposed MLD algorithm, the binary differential evolution (BDE) algorithm is used, which is a swarm intelligence technique. Computer simulations are used to evaluate the BDE algorithm's usefulness for reducing computational complexity and improving the BER of the communication system.

On deep holes of primitive projective Reed-Solomon codes

Primitive projectiveReed-Solomon code, which is an important class ofmaximum distance separable codes, is currently widely usedin digital communication. We usually usethe maximum likelihood decoding algorithm in thedecoding process of Reed-Solomon codes. For the receivedword ${\boldsymbol~u}\in\mathbb{F}_q^n$, the maximum likelihooddecoding algorithm lies in determining its error distance$d({\boldsymbol~u},C)$. It is well known that $d~({\boldsymbol~u},~C)\leq~\rho(C)$with $\rho(C)$ being the covering radius of code $C$.If $d({\boldsymbol~u},~C)=\rho(C)$, then ${\boldsymbol~u}$ is calleda deep hole of $C$. In this paper, we obtain a class of deepholes of primitive projective Reed-Solomon codes${\rm~PPRS}_q(\mathbb{F}_q^*,k)$. In fact, by usingthe generating matrix of maximaldistance separable codes over the finite field $\mathbb{F}_q$and Vandermonde determinant, we show that if $q\ge~4$, $k$ is aninteger with $2\le~k\le~q-2$ and the Lagrange interpolating polynomialof the first $q-1$ components of the received word ${\boldsymbol~u}$ is$\lambda~x^{q-2}+f_{\le{k-2}}(x)$, with $\lambda\in\mathbb{F}_q^*$and $f_{\leqslant{k-2}}(x)$ being a polynomial of degree no morethan $k-2$ over $\mathbb{F}_q$, and the $q$-th component of${\boldsymbol~u}$ is $0$, then ${\boldsymbol~u}$ is a deep hole of the primitiveprojective Reed-Solomon code ${\rm~PPRS}_q(\mathbb{F}_q^*,k)$.

Matrix embedding in steganography with binary Reed–Muller codes

This study presents a modified majority-logic decoding algorithm of Reed-Muller (RM) codes for matrix embedding (ME) in steganography. An ME algorithm uses linear block code to improve the embedding efficiency in steganography. The optimal embedding algorithm in steganography is equivalent to the maximum likelihood decoding (MLD) algorithm in error-correcting codes. The main disadvantage of ME is that the equivalent MLD algorithm of lengthy embedding codes requires highly complex embedding. This study used RM codes to embed data in binary host images. The authors propose a novel low-complexity embedding algorithm that uses a modified majority-logic algorithm to decode RM codes, in which a message-passing algorithm (i.e. sum-product, min-sum, or bias propagation) is performed on the highest order of information bits in the RM codes. The experimental results indicate that integrating bias propagation into the proposed scheme achieves superior embedding efficiency (relative to when the sum-product or min-sum algorithm is used) and can even achieve the embedding bound of RM codes.

Efficient Decoding Technique for Block Product Turbo Codes

In this paper, we propose an efficient technique for an iterative soft input soft output (SISO) decoding algorithm of block turbo codes with constituent Hamming codes. Pyndiah’s iterative decoding algorithm for block turbo codes supports various code rates by using concatenation, and shows high bit error rate (BER) performance in an additive white Gaussian noise channel (AWGN) environment as the iterative decoding progresses. However, one disadvantage of this code is that the delay increases during decoding with an increasing number of iterations. By employing syndrome-based iterative SISO decoding and iterative SISO decoding using an alpha value, we could obtain high BER performance. In addition, we propose a SISO-hybrid maximum likelihood decoding (SISO-H-MLD) algorithm which achieves almost the same performance as SISO decoding while reducing the number of iterations.

On the criteria for designing complex orthogonal space-time block codes

A complex orthogonal design (COD) used in space-time block codes is a kind of combinatorial design. It has been well studied because it has a fast maximum-likelihood decoding algorithm and achieves full diversity. When designing CODs, there are seven characteristics that should be considered, which include code rate, decoding delay, transceiver signal linearization, peak-to-average power ratio, etc. In this paper, we study the relationships among these design criteria. We prove that the maximum rate of generalized CODs (GCODs) that meet the last five criteria is 1/2. Moreover, we present a new method to construct GCODs based on CODs. Using this method on balanced complex orthogonal designs (BCODs), we obtain a novel class of GCODs that performs almost perfectly with respect to all the seven properties.

Full-diversity high-rate space-time block-coded systems using estimated channel state information for symbol detection

In this paper, the frame-error-rate (FER) performance of full-diversity high-rate space-time block-coded (STBC) system is presented for quadrature phase-shift keying (QPSK) digital modulation technique, which is using the numeric-variable-forgetting-factor (NVFF) least-squares (LS) algorithm based channel estimator. The STBC transmission data-rates with two-transmitter antennas and with four-transmitter antennas are achieved by maximizing the coding gain and by minimizing the peak-to-minimum-power-ratio while using the selective power scaling of the information symbols, which also guarantees full-diversity in case of the two-transmitter antennas. The imperfect channel estimation leads to degradation in the FER performance of the low complexity maximum likelihood decoding algorithms, though the optimum power scaling factor is incorporated along with the constellation rotation scheme. Simulation results are presented to demonstrate that the FER performance of high-rate STBC system under idealistic conditions is quite different from its performance under realistic wireless environment due to the channel estimation errors. Therefore, the tracking performance of the pilot channel estimator is found to play significant role in the overall performance evaluation of the proposed wireless systems under slowly time-varying channels also in case of the general M-ary phase-shift keying (M-PSK) constellations. Key words: Space-time block-code, coding gain, peak-to-minimum-power-ratio, numeric-variable-forgetting-factor, variable-forgetting-factor, least-squares algorithm.

Study on Quasi-Orthogonal Space Time Block Code Based on Amplify-and-Forward Relay Networks

A kind of quasi-orthogonal space time block code for amplify-and-forward relay networks is analyzed in this paper. At the source, 4 single-antenna users transmit the quasi-orthogonal space time block code matrix. At the relay node, fast maximum-likelihood decoding algorithm is adopted and the decoded symbols are then encoded into quasi-orthogonal code which is transmitted through transmit antenna. At the destination, pair-wise decoding algorithm is utilized. Finally we compare the performance of the quasi-orthogonal space time block code in relay networks with that of the quasi-orthogonal space time block code in traditional communication system. Simulation results show that at the given BER of 10-3, the new code can provide about gain of 4dB and 1 dB comparing with that for the traditional quasi-orthogonal code, respectively. And at the given BER of 10-5, the new code can provide about gain of 4.6dB and 2.6dB comparing with that for the old quasi-orthogonal code, respectively.

On-the-Fly Maximum-Likelihood Decoding of Raptor Codes over the Binary Erasure Channel

In this letter, we propose an efficient on-the-fly algorithm for maximum-likelihood decoding of Raptor codes used over the binary erasure channel. It is shown that our proposed decoding algorithm can reduce the actual elapsed decoding time by more than two-thirds with respect to an optimized conventional maximum-likelihood decoding.

Cooperative STBC with fuzzy election applied to surveillance wireless video sensor networks

Wireless video sensor networks (WVSN) have been envisioned for a wide range of important applications such as video surveillance, emergency response, environmental and habitat monitoring. Transmitting video over WVSN requires large bandwidth as well as lifetime of sensor nodes long enough to fulfill the application requirements. This paper proposes an integrated system for WVSN (mainly used for surveillance) that combines the use of adaptive cooperative diversity and fuzzy logic. The cooperative spacetime block codes (STBC) are used to achieve full diversity with a simple maximum-likelihood decoding algorithm in order to enhance the performance of the considered WVSN. The performance comparison between the proposed system and a non-cooperative scheme is presented based on network lifetime, quality of transmitted videos, and required number of retransmissions under varying propagation scenarios in a WVSN. Simulation results show better performance for the proposed system, presenting greater lifetime, higher peak signal-to-noise ratio (PSNR) values of transmitted videos and lower propagation delay.

An efficient algorithm for ML decoding of raptor codes over the binary erasure channel

In this letter, we propose an efficient algorithm for maximum-likelihood decoding of Raptor codes used over the binary erasure channel. The algorithm is inspired by the decoding scheme suggested in the 3GPP multimedia broadcast/multicast services standard and is an improved version of it.

State Transparent Convolutional Codes and Their Novel Maximum-Likelihood Decoding Algorithm

Almost all the probabilistic decoding algorithms known for convolutional codes, perform decoding without prior knowledge of the error locations. Here, we introduce a novel maximum-likelihood decoding algorithm for a new class of convolutional codes named as the state transparent convolutional (STC) codes, which due to their properties error detection and error locating is possible prior to error correction. Hence, their decoding algorithm, termed here as the STC decoder, allows an error correcting algorithm to be applied only to the erroneous portions of the received sequence referred to here as the error spans (ESPs). We further prove that the proposed decoder, which locates the ESPs and applies the Viterbi algorithm (VA) only to these portions, always yields a decoded path in trellis identical to the one generated by the Viterbi decoder (VD). Due to the fact that the STC decoder applies the VA only to the ESPs, hence percentage of the single-stage (per codeword) trellis decoding performed by the STC decoder is considerably less than the VD, which is applied to the entire received sequence and this reduction is overwhelming for the fading channels, where the erroneous codewords are mostly clustered. Furthermore, through applying the VA only to the ESPs, the resulting algorithm can be viewed as a new formulation of the VD for the STC codes that analogous to the block decoding algorithms provides a predecoding error detection and error locating capabilities, while performing less single-stage trellis decoding.

Subsidiary maximum likelihood iterative decoding based on extrinsic information

This paper proposes a multimodal generalized Gaussian distribution (MGGD) to effectively model the varying statistical properties of the extrinsic information. A subsidiary maximum likelihood decoding (MLD) algorithm is subsequently developed to dynamically select the most suitable MGGD parameters to be used in the component maximum a posteriori (MAP) decoders at each decoding iteration to derive the more reliable metrics performance enhancement. Simulation results show that, for a wide range of block lengths, the proposed approach can enhance the overall turbo decoding performance for both parallel and serially concatenated codes in additive white Gaussian noise (AWGN), Rician, and Rayleigh fading channels.

Spectrally efficient differential space-time coding using non-full-diverse constellations

In this paper, a method is proposed to construct spectral-efficient unitary space–time codes for high-rate differential communications over multiple-antenna channels. Unlike most of the known methods, which are designed to maximize the diversity product (the minimum determinant distance), our objective is to increase the spectral efficiency. The simulation results indicate that for high spectral efficiency and for more than one receive antenna, the new method significantly outperforms the existing alternatives. In the special case of two transmit antennas, which is the main focus of this paper, the relation between the proposed code and the Alamouti scheme helps us to provide an efficient maximum-likelihood decoding algorithm. Also, we demonstrate that similar ideas can be applied for designing codes for more than two transmit antennas. As an example, we present a construction for a 4 $times$ 4 unitary constellations which has a good performance, as compared with the other known codes.

Simple maximum-likelihood decoding of generalized first-order Reed-Muller codes

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).

Improved Performance of Maximum Likelihood Decoding Algorithm with Efficient Use of Algebraic Decoder

A new soft decision maximum-likelihood decoding algorithm, which generates the minimum set of candidate codewords by efficiently applying the algebraic decoder is proposed. As a result, the decoding complexity is reduced without degradation of performance. The new algorithm is tested and verified by simulation results.

Dynamic Programming in Digital Communications: Viterbi Decoding to Turbo Multiuser Detection

Many important signal processing tasks in digital communications are based on integer programming problems whose raw complexity is extremely high. Such problems include the decoding of convolutional codes, channel equalization, multiuser detection, and the joint performance of these tasks. In each of these problems, the high complexity arises from the need to perform simultaneous processing on long sequences of finite-valued symbols in order to optimally detect or decode them. Fortunately, the complexity of these optimization problems can often be greatly reduced through the use of dynamic programming, which efficiently finds optimal [e.g., maximum likelihood (ML) or maximum a posteriori probability (MAP)] decisions in long sequences of symbols. This paper reviews four decades of progress in this area: the Viterbi algorithm for ML decoding of convolutional codes of the 1960s; the ML sequence detectors for channel equalization and the BCJR algorithm for MAP decoding of convolutional codes of the 1970s; the ML and MAP multiuser detectors of the 1980s; and combinations of these through the turbo processing of the 1990s.

Soft decision decoding of Reed-Solomon codes

This paper presents a maximum-likelihood decoding (MLD) and a suboptimum decoding algorithm for Reed-Solomon (RS) codes. The proposed algorithms are based on the algebraic structure of the binary images of RS codes. Theoretical bounds on the performance are derived and shown to be consistent with simulation results. The proposed suboptimum algorithm achieves near-MLD performance with significantly lower decoding complexity. It is also shown that the proposed suboptimum, algorithm has better performance compared with generalized minimum distance decoding, while the proposed MLD algorithm has significantly lower decoding complexity than the well-known Vardy-Be'ery (1991) MLD algorithm.