Progressive Algebraic Soft-Decision Decoding of Reed–Solomon Codes Using Module Minimization

Reed-Solomon (RS) codes are used as error-correcting codes in numerous digital communication and storage systems. Algebraic soft-decision decoding (ASD) of RS codes can achieve substantial coding gain with polynomial complexity. Among practical ASD algorithms, the iterative bit-level generalized minimum distance (BGMD) decoding can achieve similar or higher coding gain with lower complexity. The interpolation is a major step of ASD. The maximum achievable speed of this step is limited by the inherent serial nature of the interpolation algorithm. In this paper, a novel interpolation scheme that is capable of combining multiple interpolation iterations, as well as sharing interpolation results from previous decoding iterations, is developed for the iterative BGMD decoding. In addition, efficient VLSI architectures are proposed to implement the developed scheme. Based on the proposed architectures, an interpolator for a (255, 239) RS code is implemented on field programmable gate array (FPGA) devices. On a Xilinx Virtex-II device, our interpolator can achieve a throughput of 440 Mbps, which is 64% higher than the fastest previous design, with 51% less FPGA resource.

Reed-Solomon (RS) codes are widely used in digital communication and storage systems. Algebraic soft-decision decoding (ASD) of RS codes can obtain significant coding gain over the hard-decision decoding (HDD). Compared with other ASD algorithms, the low-complexity Chase (LCC) decoding algorithm needs less computation complexity with similar or higher coding gain. Besides employing complicated interpolation algorithm, the LCC decoding can also be implemented based on the HDD. However, the previous syndrome computation for 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">η</sup> test vectors and the key equation solver (KES) in the HDD requires long latency and remarkable hardware. In this brief, a unified syndrome computation algorithm and the corresponding architecture are proposed. Cooperating with the KES in the reduced inversion-free Berlekamp-Messy algorithm, the reduced-complexity LCC RS decoder can speed up by 57% and the area will be reduced to 62% compared with the original design for η = 3.

We propose a Reed-Solomon (RS) code construction of the 1-out-n (ring) signature scheme. It is obtained from the observation of the equivalency between the erasure correction technique of the RS code and the polynomial interpolation. The structure is very simple and yields a ring equation that can appropriately denoted by Z 1 + ... + Z n =v, which represents the summation of n evaluations of a polynomial. We also show how to extend the generic RST scheme [6] to a t-out-n threshold ring signature scheme.

Determining deep holes is an important topic in decoding Reed-Solomon codes.\nIn a previous paper [8], we showed that the received word $u$ is a deep hole of\nthe standard Reed-Solomon codes $[q-1, k]_q$ if its Lagrange interpolation\npolynomial is the sum of monomial of degree $q-2$ and a polynomial of degree at\nmost $k-1$. In this paper, we extend this result by giving a new class of deep\nholes of the generalized Reed-Solomon codes.\n

Problem statement. Deep neural networks are successfully used in channel coding to improve the quality of decoding. However, modern neural channel decoders cannot achieve high decoding performance and low complexity at the same time. To overcome this problem, it is necessary to develop a decoder with a double residual neuron. By integrating residual input and residual learning into the neural channel decoder architecture, it will significantly improve decoding performance while maintaining low complexity. Goal. Development of a decoder architecture with a dual residual neural network, which provides high decoding performance in channel coding with low model complexity and computational costs. Results. The simulation results show that the proposed decoder with a double residual neural network provides a significant increase in decoding performance. Compared to a neural decoder based on a trust propagation algorithm, a decoder with a dual residual neural network provides an additional 0.5-1.8 dB encoding gain compared to different channel codes. Compared to the hypergraph neural decoder, which has the best error correction capability among all existing neural channel decoders, the dual residual neural network decoder also provides similar or even better decoding performance for different channel codes. Practical significance. The results obtained can be used to improve decoding efficiency and improve performance in channel coding. The work was carried out with the financial support of the Ministry of Science and Higher Education of the Russian Federation within the framework of the state assignment "youth laboratory" No. FZGM-2024-0003.

This paper presents a comprehensive analysis and implementation of various error correction coding methods, focusing on Reed-Solomon (RS) codes and Liberation codes in RAID6 systems. RS codes are emphasized for their robust error correction capabilities, making them ideal for high-reliability environments such as enterprise data centers and mission-critical systems, where the ability to correct multiple errors or data losses is crucial. In contrast, Liberation codes are noted for their superior encoding efficiency and lower computational complexity, making them well-suited for high-performance systems with stringent speed requirements. Through experimental comparisons, this study also evaluates Cauchy Reed-Solomon (CRS) codes, EVENODD codes, and Blaum-Roth (BR) codes, identifying the strengths and weaknesses of each method in various scenarios. The findings underscore the importance of selecting the appropriate coding technique based on system-specific requirements, as different use cases benefit from different error correction schemes in terms of performance and data reliability. Additionally, this research highlights the potential of these coding methods in broader distributed storage applications and discusses the trade-offs between them. Future research directions are suggested, particularly in optimizing existing codes and expanding their application in more complex and distributed storage environments.

Color-image data is generally more complex than black and whilt version in terms of the number of variables in the channels and “color constancy” issue. Most of two-dimensional (2D) codes are dominant in a black and white domain such as QR code, PDF417, Data Matrix and Maxicode rather than in color domain, affected usually by such technical difficulities of handing color variables. Conventional 2D codes adopt Reed-Solomon (RS) algorithm for their error control codes mainly to recover from damages such as stains, scrathes and spots, while highlight and shadow issues in color codes are much more complex. In this paper we propose a new decoding approach by applying recoverable erasures by RS algorithm in color codes to resolve color recognition issue effectively. Using erasure concept ultimately for color constantcy, marking erasure strategy during color recognition processing steps is very crucial. Our algorithm ultimately mitigates color recognition load overall in decoding color codes. Consequently our new erasure decision algorithm prevents color recognition failures within erasure correction capability of RS algorithm, and it can be applied along with other color constancy algorithms easily to increase overall decoding performance just using RS erasure.

Multiple description coding has been receiving attention as a robust transmission framework for multimedia services. This paper studies the iterative decoding of FEC-based multiple description codes. The proposed decoding algorithms take advantage of the error detection capability of Reed-Solomon (RS) erasure codes. The information of correctly decoded RS codewords is exploited to enhance the error correction capability of the Viterbi algorithm at the next iteration of decoding. In the proposed algorithm, an intradescription interleaver is synergistically combined with the iterative decoder. The interleaver does not affect the performance of noniterative decoding but greatly enhances the performance when the system is iteratively decoded. We also address the optimal allocation of RS parity symbols for unequal error protection. For the optimal allocation in iterative decoding, we derive mathematical equations from which the probability distributions of description erasures can be generated in a simple way. The performance of the algorithm is evaluated over an orthogonal frequency-division multiplexing system. The results show that the performance of the multiple description codes is significantly enhanced.

Algebraic soft-decision decoding of Reed-Solomon (RS) codes delivers promising coding gains over conventional hard-decision decoding. The most computationally demanding step in the soft-decision decoding is bivariate polynomial interpolation. In this paper, we present a very efficient high speed interpolation architecture based on hybrid data representation. It is shown that the proposed architecture is inherently scalable and can be extensively pipelined to achieve very high clock speed. By further incorporating the maximum overlapping for computations at adjacent iterations, the proposed architecture demonstrates significant advantages over conventional designs. It is estimated that over 1 Gbps data rate can be achieved using the presented work with moderate complexity.

Algebraic soft-decision decoding (ASD) of Reed-Solomon (RS) codes can provide substantial coding gain with polynomial complexity. Among the ASD algorithms with practical multiplicity assignment schemes, the Low-Complexity Chase (LCC) decoding can achieve similar or higher coding gain. Interpolation is a major step in ASD. Since the maximum multiplicity of the interpolation point is only one in LCC, the interpolation over each point has low complexity. However, 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">η</sup> test vectors are involved in the LCC, and the interpolation needs to be carried out on each of them. In order to reduce the computational complexity of the overall interpolation, intermediate results can be stored and shared. Nevertheless, the storage requires large memory, which accounts for a significant portion of the overall hardware requirement of the interpolator. In this paper, we propose a novel interpolation procedure, in which the 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">η</sup> test vectors are mapped to the vertices of a dimension-η hypercube and the vectors mapped to adjacent vertices have only one different entry. In addition, a backward interpolation is proposed to support the traversal from one vertex to its neighbors. Traveling through the entire hypercube, the interpolation over each test vector can be done one after another and the memory requirement is reduced by a factor of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">η-1</sup> . Efficient architectures are also developed for the proposed interpolation procedure. With about the same latency and the same number of gates as in prior efforts, our architecture can reduce the memory size to 25% and the number of registers to 57% for a (255, 239) RS code with η = 3. The saving further increases with η.

Rateless802.11: Extending WiFi applicability in extremely poor channels

Highly reliable data transmission using Reed-Solomon(RS)/convolutional concatenated coding in conjunction with hybrid type-I automatic retransmission request (ARQ) is investigated. The study is based on an analysis of the tradeoff between error correction and error detection capabilities of RS codes for the direct-sequence code-division multiple-access (DS-CDMA) systems. Two RS decoding schemes are considered: error-only decoding (decoding without side information) and error-and-erasure (EE) decoding. In the EE scheme, the inner code is decoded with a modified Viterbi algorithm, which produces reliable information along with the decoded output. Numerical results illustrate the importance of properly selecting the design parameters for the purpose of maximizing spectral efficiency. The results also show that an E/sub b//N/sub 0/ gain of 0.1 to 0.5 dB can be obtained by using EE decoding, depending on the selected parameters.

Reed-Solomon (204,188) code has been widely used in many digital multimedia broadcasting systems. This paper focuses on the low complexity derivation of adaptive belief propagation bit-level soft decision decoding for this code. Simulation results demonstrate that proposed normalized min-sum algorithm as belief propagation (BP) process provides the same decoding performance in terms of packet-error-rate as sum-product algorithm. An outer adaptation scheme by moving adaptation process of parity check matrix out of the BP iteration loop is also introduced to reduce decoding complexity. Simulation results show that the proposed two schemes perform a good trade-off between the decoding performance and decoding complexity.KeywordsReed-Solomon codesAdaptive belief propagationSum-Product algorithmMin-sum algorithmBit-level parity check matrixGaussian elimination

In this paper, we impose a few sparse parity-check (PC) equations to a Reed-Solomon (RS) code as a concatenation of PC and RS codes, which can provide considerable performance gain by an efficient soft-decision decoding for some existing RS coded systems, such as satellite communications. We propose two construction methods for the sub-matrices. The type-1 construction method directly concatenates a specially designed sub-matrix with regular row weight, and the type-2 construction method is developed to achieve a much lower encoding complexity while maintaining the performance gain. With the help of PC concatenation, the quantity of the channel soft information can be refined by the newly added sparse PC equations which significantly improve the performance of Chase-type decoders. In order to reduce the decoding complexity, we propose a threshold-based noise-aided Chase algorithm (TB-NACA) for PC-RS codes, which achieves significant performance gain compared to the hard -decision decoding of the original RS codes. Moreover, the proposed PC-RS code under the TB-NACA decoding can even significantly outperform the conventional Chase-type decoders over the AWGN channels and Rayleigh fading channels with high-order QAM.

A new iterative soft decision decoding method for Reed-Solomon (RS) codes is proposed. This method is based on bit level belief propagation (BP) decoding. In order to make BP decoding effective for RS codes, we use an extended binary parity check matrix with a lower density and reduced number of 4-cycles compared to the original binary parity check matrix of the code. In our proposed method, we take advantage of the cyclic structure of RS codes. Based on this property, we can apply the belief propagation algorithm on any cyclically shifted version of the received symbols with the same binary parity check matrix. For each shifted version of received symbols, the geometry of the factor graph will change and deterministic errors can be avoided. Our method results in considerable performance improvement of RS codes compared to hard decision decoding. The performance is also superior to some popular soft decision decoding methods.