Efficient Decoding Algorithm Research Articles

Turbo decoding of concatenated codes based on RS codes using adapted scaling factors

Iteratively decoded block turbo codes are product codes that exhibit excellent performance with reasonable complexity. In this paper, a generalization of parallel concatenated block codes (GPCBs) based on RS codes is presented. We propose an efficient decoding algorithm with modifications of the Chase-Pyndiah algorithm is written by using Weighting factor α and Reliability factor β. In this work, we studied the effect of diverse parametres such as the effect of various component codes, interleaver size (number of sub-blocks) and number of iterations. The simulation results shows the relevance of the adapted parameters to decode generalized parallel concatenated block codes based on RS codes. The proposed algorithm (MCP) using the adapted parameters performs better than the one using with empirical parameters (CP).

Effective decoders for DNA codes

A number of applications use DNA as a storage mechanism. Because processes in these applications may cause errors in the data, the information must be encoded as one of a chosen set of words that are well separated from one another — a DNA error-correcting code. Typically, the types of errors that may occur include insertions, deletions and substitutions of symbols, making the edit metric the most suitable choice to measure the distance between strings. Decoding, the process of recovering the original word when errors occur, is complicated by biological restrictions combined with a high cost to calculate edit distance.Side effect machines (SEMs), an extension of finite state machines, can provide efficient decoding algorithms for such codes. Several codes of varying lengths are used to study the effectiveness of evolutionary programming (EP) as a general approach for finding SEMs for edit metric decoding. Two classification methods (direct and fuzzy classification) are compared, and different EP settings are examined to observe how decoding accuracy is affected. Regardless of code length, the best results are found using fuzzy classification. The best accuracy is seen for codes of length 10, for which a maximum accuracy of up to 99.4% is achieved for distance 1 and distance 2 and 3 achieve up to 97.1% and 85.9%, respectively. Additionally, the SEMs are examined for potential bloat by comparing the number of reachable states against the total number of states. Bloat is seen more in larger machines than in smaller machines. Furthermore, the results are analysed to find potential trends and relationships among the parameters, with the most consistent trend being that, when allowed, the longer codes generally show a propensity for larger machines.

Decoding Reed–Solomon Skew-Differential Codes

A large class of MDS linear codes is constructed. These codes are endowed with an efficient decoding algorithm. Both the definition of the codes and the design of their decoding algorithm only require from Linear Algebra methods, making them fully accessible for everyone. Thus, the first part of the paper develops a direct presentation of the codes by means of parity-check matrices, and the decoding algorithm rests upon matrix and linear maps manipulations. The somewhat more sophisticated mathematical context (non-commutative rings) needed for the proof of the correctness of the decoding algorithm is postponed to the second part. A final section locates the Reed-Solomon skew-differential codes introduced here within the general context of codes defined by means of skew polynomial rings.

Decoding of Interleaved Alternant Codes

Interleaved Reed–Solomon codes admit efficient decoding algorithms which correct burst errors far beyond half the minimum distance in the random errors regime, e.g., by computing a common solution to the Key Equation for each Reed–Solomon code, as described by Schmidt et al. If this decoder does not succeed, it may either <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">fail</i> to return a codeword or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">miscorrect</i> to an incorrect codeword, and good upper bounds on the fraction of error matrices for which these events occur are known. The decoding algorithm immediately applies to interleaved alternant codes as well, i.e., the subfield subcodes of interleaved Reed–Solomon codes, but the fraction of decodable error matrices differs, since the error is now restricted to a subfield. In this paper, we present new general lower and upper bounds on the fraction of error matrices decodable by Schmidt et al.’s decoding algorithm, thereby making it the only decoding algorithm for interleaved alternant codes for which such bounds are known.

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].

Cardinal rank metric codes over Galois rings

In 1985, Gabidulin introduced the rank metric in coding theory over finite fields, and used this kind of codes in a McEliece cryptosystem, six years later. In this paper, we consider rank metric codes over Galois rings. We propose a suitable metric for codes over such rings, and show its main properties. With this metric, we define Gabidulin codes over Galois rings, propose an efficient decoding algorithm for them, and hint their cryptographic application.

Probability-Based Ordered-Statistics Decoding for Short Block Codes

This letter proposes an efficient probability-based ordered-statistics decoding (PB-OSD) algorithm for short block-length codes. In PB-OSD, we derive two probabilistic measures on the codeword estimates and test error patterns, respectively referred to as the success probability and promising probability. Based on these probabilities, a stopping criterion and a discarding criterion are developed to reduce the number of test error patterns and limit the decoding complexity. To further reduce the complexity, we propose a tree-based search strategy to find the most likely test error patterns in reprocessing stage of the OSD algorithm. Simulation results show that PB-OSD significantly reduces the decoding complexity under the same error performance, compared to the original OSD algorithm.

A Novel Niederreiter-like cryptosystem based on the (u|u+υ)-construction codes

In this paper, we present a new variant of the Niederreiter Public Key Encryption (PKE) scheme which is resistant against recent attacks. The security is based on the hardness of the Rank Syndrome Decoding (RSD) problem and it presents a (u|u+υ)-construction code using two different types of codes: Ideal Low Rank Parity Check (ILRPC) codes andλ-Gabidulin codes. The proposed encryption scheme benefits are a larger minimum distance, a new efficient decoding algorithm and a smaller ciphertext and public key size compared to the Loidreau’s variants and to its IND-CCA secure version.

Reduced‐Complexity LDPC Decoding for Next‐Generation IoT Networks

Low‐density parity‐check (LDPC) codes have become the focal choice for next‐generation Internet of things (IoT) networks. This correspondence proposes an efficient decoding algorithm, dual min‐sum (DMS), to estimate the first two minima from a set of variable nodes for check‐node update (CNU) operation of min‐sum (MS) LDPC decoder. The proposed architecture entirely eliminates the large‐sized multiplexing system of sorting‐based architecture which results in a prominent decrement in hardware complexity and critical delay. Specifically, the DMS architecture eliminates a large number of comparators and multiplexors while keeping the critical delay equal to the most delay‐efficient tree‐based architecture. Based on experimental results, if the number of inputs is equal to 64, the proposed architecture saves 69%, 68%, and 52% area over the sorting‐based, the tree‐based, and the low‐complexity tree‐based architectures, respectively. Furthermore, the simulation results show that the proposed approach provides an excellent error‐correction performance in terms of bit error rate (BER) and block error rate (BLER) over an additive white Gaussian noise (AWGN) channel.

Constant overhead quantum fault tolerance with quantum expander codes

The threshold theorem is a seminal result in the field of quantum computing asserting that arbitrarily long quantum computations can be performed on a faulty quantum computer provided that the noise level is below some constant threshold. This remarkable result comes at the price of increasing the number of qubits (quantum bits) by a large factor that scales polylogarithmically with the size of the quantum computation we wish to realize. Minimizing the space overhead for fault-tolerant quantum computation is a pressing challenge that is crucial to benefit from the computational potential of quantum devices. In this paper, we study the asymptotic scaling of the space overhead needed for fault-tolerant quantum computation. We show that the polylogarithmic factor in the standard threshold theorem is in fact not needed and that there is a fault-tolerant construction that uses a number of qubits that is only a constant factor more than the number of qubits of the ideal computation. This result was conjectured by Gottesman who suggested to replace the concatenated codes from the standard threshold theorem by quantum error-correcting codes with a constant encoding rate. The main challenge was then to find an appropriate family of quantum codes together with an efficient classical decoding algorithm working even with a noisy syndrome. The efficiency constraint is crucial here: bear in mind that qubits are inherently noisy and that faults keep accumulating during the decoding process. The role of the decoder is therefore to keep the number of errors under control during the whole computation. On a technical level, our main contribution is the analysis of the SMALL-SET-FLIP decoding algorithm applied to the family of quantum expander codes . We show that it can be parallelized to run in constant time while correcting sufficiently many errors on both the qubits and the syndrome to keep the error under control. These tools can be seen as a quantum generalization of the BIT-FLIP algorithm applied to the (classical) expander codes of Sipser and Spielman.

A New Construction of Nonlinear Codes via Rational Function Fields

It is well known that constructing codes with good parameters is one of the most important and fundamental problems in coding theory. Though a great many of good codes have been produced, most of them are defined over alphabets of sizes equal to prime powers. In this article, we provide a new explicit construction of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(q+1)$ </tex-math></inline-formula> -ary nonlinear codes via rational function fields, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> is a prime power. Our codes are constructed by evaluations of rational functions at all rational places (including the place of “infinity”) of the rational function field. Compared to the rational algebraic geometry codes, the main difference is that we allow rational functions to be evaluated at pole places. After evaluating rational functions from a union of Riemann-Roch spaces, we obtain a family of nonlinear codes with length <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q+1$ </tex-math></inline-formula> over the alphabet <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{q}\cup \{\infty \}$ </tex-math></inline-formula> . As a result, our codes have reasonable parameters as they are rather close to the Singleton bound. Furthermore, our codes have better parameters than those obtained from MDS codes via code alphabet restriction or extension. Amazingly, an efficient decoding algorithm can be provided for our codes.

Decoding Kinematic Information From Primary Motor Cortex Ensemble Activities Using a Deep Canonical Correlation Analysis.

The control of arm movements through intracortical brain–machine interfaces (BMIs) mainly relies on the activities of the primary motor cortex (M1) neurons and mathematical models that decode their activities. Recent research on decoding process attempts to not only improve the performance but also simultaneously understand neural and behavioral relationships. In this study, we propose an efficient decoding algorithm using a deep canonical correlation analysis (DCCA), which maximizes correlations between canonical variables with the non-linear approximation of mappings from neuronal to canonical variables via deep learning. We investigate the effectiveness of using DCCA for finding a relationship between M1 activities and kinematic information when non-human primates performed a reaching task with one arm. Then, we examine whether using neural activity representations from DCCA improves the decoding performance through linear and non-linear decoders: a linear Kalman filter (LKF) and a long short-term memory in recurrent neural networks (LSTM-RNN). We found that neural representations of M1 activities estimated by DCCA resulted in more accurate decoding of velocity than those estimated by linear canonical correlation analysis, principal component analysis, factor analysis, and linear dynamical system. Decoding with DCCA yielded better performance than decoding the original FRs using LSTM-RNN (6.6 and 16.0% improvement on average for each velocity and position, respectively; Wilcoxon rank sum test, p < 0.05). Thus, DCCA can identify the kinematics-related canonical variables of M1 activities, thus improving the decoding performance. Our results may help advance the design of decoding models for intracortical BMIs.

Recursive Projection-Aggregation Decoding of Reed-Muller Codes

We propose a new class of efficient decoding algorithms for Reed-Muller (RM) codes over binary-input memoryless channels. The algorithms are based on projecting the code on its cosets, recursively decoding the projected codes (which are lower-order RM codes), and aggregating the reconstructions (e.g., using majority votes). We further provide extensions of the algorithms using list-decoding. We run our algorithm for AWGN channels and Binary Symmetric Channels at the short code length (≤ 1024) regime for a wide range of code rates. Simulation results show that in both low code rate and high code rate regimes, the new algorithm outperforms the widely used decoder for polar codes (SCL+CRC) with the same parameters. The performance of the new algorithm for RM codes in those regimes is in fact close to that of the maximal likelihood decoder. Finally, the new decoder naturally allows for parallel implementations.

Multi-objective scheduling in hybrid flow shop: Evolutionary algorithms using multi-decoding framework

Hybrid flow shops are common manufacturing environments applied in many industrial fields. This paper tackles the scheduling problem in hybrid flow shop with unrelated machines, machine eligibility and sequence-dependent setup times (SDST) to minimize the bi-criteria of total tardiness and total setup time. Evolutionary algorithms (EAs) are adopted to solve the problem. Firstly, four efficient decoding algorithms using different machine selection rules are developed for constructing a schedule from a job permutation. These decoding algorithms are able to map the job permutation space to distinct regions in the objective space. Then, we propose a multi-decoding framework (MDF) for taking advantage of multiple decoding algorithms along one evolution path. The hybridization of MDF and EAs leads to a hyper-heuristic approach. The proposed MDF is coupled with a genetic algorithm to solve the problem in “a priori” approach, that is, to optimize a convex combination of the objectives given user preference information. The framework is also embedded to a multi-objective genetic algorithm, known as NSGA-II, to solve the problem in “a posteriori” approach, which aims at approximating the Pareto-optimal set for the user to make posterior decisions. The efficiency of the proposed methods is validated by numerical results. More specifically, when “a priori” approach is used, the proposed MDF helps EAs adjusting the adopted decoding scheme and generating solution aligned to the user preference; when “a posteriori” approach is applied, the MDF extends the search space and improves the solution quality.

Decoding Binary Linear Codes Over Channels With Synchronization Errors

Time synchronization is crucial for the safe and reliable operation of the fifth generation (5G) network, especially for applications requiring ultra-reliable low-latency data transmissions. The time synchronization problem, however, becomes increasingly challenging in high-mobility scenarios because the channel conditions, e.g., the multipath delay spread may vary rapidly. While there exist numerous works on the design of efficient channel decoding algorithms, decoding linear codes such as polar codes in the presence of synchronization errors is a less-explored topic. In this paper, we aim to fill this void and develop a systemic approach to decode general binary linear codes over binary symmetric channels with synchronization errors in which the lack of synchronization is modeled as the deletion channel model. The maximum likelihood (ML) decoding problem for binary linear codes over deletion channels is first formulated as a nonlinear optimization problem, in which a set of linear constraints are employed to characterize the input-output relationship of a deletion channel. It turns out that both the objective function and the constraints of this optimization problem are nonlinear, which poses significant challenges against the design of efficient decoding algorithms. As a remedy, we first replace the nonlinear objective function of this optimization problem via a lower bound. And we prove this lower bound is a linear function in the special case that the input is binary. We then apply the linear programming (LP) relaxation approach to obtain an approximate solution to the proposed nonlinear optimization problem. An adaptive branch-and-cut decoding algorithm has also been developed by making use of the ML-certificate property of the LP decoder for deletion channel. It is seen through simulation studies that the proposed decoding algorithm can achieve close-to-optimal bit error rate (BER) decoding performance at moderate computational complexity.

Double and Triple Node-Erasure-Correcting Codes Over Complete Graphs

In this paper we study array-based codes over graphs for correcting multiple node failures. These codes have applications to neural networks, associative memories, and distributed storage systems. We assume that the information is stored on the edges of a complete undirected graph and a node failure is the event where all the edges in the neighborhood of a given node have been erased. A code over graphs is called ρ-node-erasure-correcting if it allows to reconstruct the erased edges upon the failure of any ρ nodes or less. We present a binary optimal construction for double-node-erasure correction together with an efficient decoding algorithm, when the number of nodes is a prime number. Furthermore, we extend this construction for triple-node-erasure-correcting codes when the number of nodes is a prime number and two is a primitive element in ℤ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> . These codes are at most a single bit away from optimality.

On quaternary Goppa codes

In Goppa (1970), V. D. Goppa presents a new family of linear codes over finite fields. This family, now known as Goppa codes, has been studied for nearly 50 years, including efficient decoding algorithms (Patterson, 1975) and its applications in cryptography (McEliece, 1978). Moreover, in Andrade et al. (2005), de Andrade and Palazzo present a generalization of Goppa codes to finite rings. Motivated by cryptographic applications, in this article we focus on Goppa codes over Z∕4Z and study some of their properties.

Joint Viterbi detection and decoding algorithm for bluetooth low energy systems

An efficient joint Viterbi detection and decoding (JVDD) algorithm for a bluetooth low energy (BLE) system is proposed. Since the convolutional coded Gaussian minimum-shift keying (GMSK) signal is specified in the BLE 5.0 standard, two Viterbi processors are needed for detection and decoding. However, the proposed JVDD scheme uses only one Viterbi processor by modifying the branch metric with inter-symbol interference information from GMSK modulation; therefore, the hardware complexity can be significantly reduced without performance degradation.

Códigos de Reed-Muller

The Reed-Muller codes were discovered by David Eugene Muller and decoded by Irving Stoy Reed in 1954. Such codes belong to the linear code family and are widely used nowadays, mainly for their simple and efficient decoding algorithm. There are several ways to define Reed-Muller codes. In this work, we present, in a clear and simple way, a recursive definition for all Reed-Muller codes of order r \in N, denoted by R (r, m), where 0 &lt;= r &lt;= m, m \in N. Using this definition, we show the main parameters: length, number of elements and minimum distance of first-order Reed-Muller codes, R(1, m) for all m \in N. In addition, we present an application of the first-order codes in a National Aeronautics and Space Administration (NASA) space program.

Projection Decoding of Some Binary Optimal Linear Codes of Lengths 36 and 40

Practically good error-correcting codes should have good parameters and efficient decoding algorithms. Some algebraically defined good codes, such as cyclic codes, Reed–Solomon codes, and Reed–Muller codes, have nice decoding algorithms. However, many optimal linear codes do not have an efficient decoding algorithm except for the general syndrome decoding which requires a lot of memory. Therefore, a natural question to ask is which optimal linear codes have an efficient decoding. We show that two binary optimal [ 36 , 19 , 8 ] linear codes and two binary optimal [ 40 , 22 , 8 ] codes have an efficient decoding algorithm. There was no known efficient decoding algorithm for the binary optimal [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes. We project them onto the much shorter length linear [ 9 , 5 , 4 ] and [ 10 , 6 , 4 ] codes over G F ( 4 ) , respectively. This decoding algorithm, called projection decoding, can correct errors of weight up to 3. These [ 36 , 19 , 8 ] and [ 40 , 22 , 8 ] codes respectively have more codewords than any optimal self-dual [ 36 , 18 , 8 ] and [ 40 , 20 , 8 ] codes for given length and minimum weight, implying that these codes are more practical.