Reed-Solomon Research Articles

Interpolation-based decoding of folded variants of linearized and skew Reed–Solomon codes

The sum-rank metric is a hybrid between the Hamming metric and the rank metric and suitable for error correction in multishot network coding and distributed storage as well as for the design of quantum-resistant cryptosystems. In this work, we consider the construction and decoding of folded linearized Reed–Solomon (FLRS) codes, which are shown to be maximum sum-rank distance (MSRD) for appropriate parameter choices. We derive an efficient interpolation-based decoding algorithm for FLRS codes that can be used as a list decoder or as a probabilistic unique decoder. The proposed decoding scheme can correct sum-rank errors beyond the unique decoding radius with a computational complexity that is quadratic in the length of the unfolded code. We show how the error-correction capability can be optimized for high-rate codes by an alternative choice of interpolation points. We derive a heuristic upper bound on the decoding failure probability of the probabilistic unique decoder and verify its tightness by Monte Carlo simulations. Further, we study the construction and decoding of folded skew Reed-Solomon codes in the skew metric. Up to our knowledge, FLRS codes are the first MSRD codes with different block sizes that come along with an efficient decoding algorithm.

Integer codes correcting single errors and detecting burst errors within two bytes

AbstractThe need for fast communication has led to the introduction of 100 Gbps optics in local area networks (LANs). To avoid retransmission of corrupted packets, in some LANs, the CRC codes were replaced by Reed‐Solomon (RS) codes. Although this solution has met its primary goal, it has two drawbacks: first, the payload size was significantly reduced, and second, each network node had to be equipped with much more complex hardware, which increased the cost of the network. Starting from the fact that the vast majority of corrupted packets contain only single errors, in this paper, a simple strategy for error control in 100 Gbps optical LANs is presented. More precisely, instead of protecting the data with the CRC or RS codes, using integer codes is suggested that can correct single errors and detect two types of burst errors: l‐bit burst errors within one b‐bit byte and h‐bit burst errors within two b‐bit bytes (1 ≤ h < l < b). Unlike the CRC or RS codes, the proposed ones use integer and lookup table operations, which are supported by all processors. In addition, the proposed codes can be interleaved without delay and without using any additional hardware. This feature allows the decoder not only to process larger data sets in parallel, but also to detect all double burst errors.

Reed Solomon Codes Against Adversarial Insertions and Deletions

In this work, we study the performance of Reed–Solomon codes against adversarial insertion-deletion (insdel) errors. We prove that over fields of size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n^O(k)</i> there are [ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n, k</i> ] Reed-Solomon codes that can decode from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> – 2 <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> + 1 insdel errors and hence attain the half-Singleton bound. We also give a deterministic construction of such codes over much larger fields (of size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n^{k^O(k)}</i> ). Nevertheless, for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> = <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> (log <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> / log log <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> ) our construction runs in polynomial time. For the special case <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k</i> = 2, which received a lot of attention in the literature, we construct an [ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> , 2] Reed-Solomon code over a field of size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">O</i> ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> ) that can decode from <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> – 3 insdel errors. Earlier constructions required an exponential field size. Lastly, we prove that any such construction requires a field of size Ω( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> ).

Symbol-level iterative information set decoding of RS codes

This paper presents the implementation of a low-complex iterative symbol-level decoding scheme for Reed-Solomon codes. Most soft-decision iterative decoders for Reed-Solomon codes work on a bit level due to the efficient passing of soft information due to the sparsity of the binary parity check matrix. These decoders yield a good error correction performance, but this comes at the cost of an increased computational complexity resulting from working at a bit-level. This study aims to lower the computational complexity of iterative decoding of Reed-Solomon codes by introducing a high-performance soft-decision iterative Reed-Solomon decoder that works on a symbol-level, in contrast to the bit-level implementation used in most iterative Reed-Solomon decoders. The proposed algorithm utilises soft information to decode while applying information set decoding techniques to an extended parity check matrix. The use of the extended parity check matrix provides additional parity check equations which assist the algorithm in determining the correct information set of symbols used in the decoding process. For a given (n,k) Reed-Solomon code, the algorithm converges to a valid codeword by correctly decoding a specified information set of k symbols from the received vector. Simulations run show the proposed algorithm performs favourably when compared to other symbol-level iterative decoders while working at a relatively lower complexity.

New MDS self-dual codes from GRS codes and extended GRS codes

Maximum-distance-separable (MDS) codes and self-dual codes are important families of classical codes in coding theory. It is of interest to investigate MDS self-dual codes. The existence of MDS self-dual codes over finite field Fq has been completely solved for q is even. In this paper, for finite field with odd characteristic, we obtain some new results on the existence of MDS self-dual codes by generalized Reed-Solomon (GRS) codes and extended GRS codes.

Implications of Power Allocation Factors on the Performance of Ergodic Sum Rate in Cooperative Device-to-Device Systems with NOMA

The power allocation issue for cooperative device-to-device (D2D) systems with non-orthogonal multiple access (NOMA) is examined in this paper. In cooperative device-to-device (D2D) NOMA systems, communication occurs in two phases: the direct transmission slot, or the path from the base station to user destinations, and the relaying slot, or the path from the relay (near the user), with different power allocation factors to improve the ergodic sum rate. It is suggested that power allocation considerations affect how well the ergodic sum rate performs with different decoding techniques. The decoding techniques include Reed Solomon's (RS) encoded single signal decoding technique and Reed Solomon's (RS) encoded maximum ratio combining (MRC) decoding technique. The error probability of successive decoding will increase if any errors occur during the successive interference cancellation (SIC) procedure at any user. In order to detect and correct multiple symbol errors in a Rayleigh fading environment, RS codes are used in the system architecture for error detection and correction. Through the results of the simulation, it is demonstrated that the proposed approach performs much better than the existing work for the fixed values of the power allocation parameters.

On the error-correcting pair for MDS linear codes with even minimum distance

The error-correcting pair is a general algebraic decoding method for linear codes, which exists for many classical linear codes. Since every linear code is contained in an MDS linear code with the same minimum distance over some finite field extension, we focus our study on MDS linear codes. It is well-known that an MDS linear code with minimum distance 2ℓ+1 has an ℓ-error-correcting pair if and only if it is a generalized Reed-Solomon code. In this paper, we show that for an MDS linear code C with minimal distance 2ℓ+2, if it has an ℓ-error-correcting pair, then the parameters of the pair are three cases. For one case, we give a necessary condition that C is a generalized Reed-Solomon code, and then give some counterexamples that C is a non-generalized Reed-Solomon code for the other two cases.

Cloud Data Recovery and Secure Data Storage with Advanced Cauchy Reed-Solomon Codes and Role-Based Cryptographic Access

Cloud Data Recovery and Secure Data Storage with Advanced Cauchy Reed-Solomon Codes and Role-Based Cryptographic Access

Cache-Based Matrix Technology for Efficient Write and Recovery in Erasure Coding Distributed File Systems

With the development of various information and communication technologies, the amount of big data has increased, and distributed file systems have emerged to store them stably. The replication technique divides the original data into blocks and writes them on multiple servers for redundancy and fault tolerance. However, there is a symmetrical space efficiency problem that arises from the need to store blocks larger than the original data. When storing data, the Erasure Coding (EC) technique generates parity blocks through encoding calculations and writes them separately on each server for fault tolerance and data recovery purposes. Even if a specific server fails, original data can still be recovered through decoding calculations using the parity blocks stored on the remaining servers. However, matrices generated during encoding and decoding are redundantly generated during data writing and recovery, which leads to unnecessary overhead in distributed file systems. This paper proposes a cache-based matrix technique that uploads the matrices generated during encoding and decoding to cache memory and reuses them, rather than generating new matrices each time encoding or decoding occurs. The design of the cache memory applies the Weighting Size and Cost Replacement Policy (WSCRP) algorithm to efficiently upload and reuse matrices to cache memory using parameters known as weights and costs. Furthermore, the cache memory table can be managed efficiently because the weight–cost model sorts and updates matrices using specific parameters, which reduces replacement cost. The experiment utilized the Hadoop Distributed File System (HDFS) as the distributed file system, and the EC volume was composed of Reed–Solomon code with parameters (6, 3). As a result of the experiment, it was possible to reduce the write, read, and recovery times associated with encoding and decoding. In particular, for up to three node failures, systems using WSCRP were able to reduce recovery time by about 30 s compared to regular HDFS systems.

List-Decoding and List-Recovery of Reed–Solomon Codes Beyond the Johnson Radius for Every Rate

Understanding the limits of list-decoding and list-recovery of Reed-Solomon (RS) codes is of prime interest in coding theory and has attracted a lot of attention in recent decades. However, the best possible parameters for these problems are still unknown, and in this paper, we take a step in this direction. We show the existence of RS codes that are list-decodable or list-recoverable beyond the Johnson radius for <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">every</i> rate, with a polynomial field size in the block length. In particular, we show that for every <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\epsilon \in (0,1)$ </tex-math></inline-formula> there exist RS codes that are list-decodable from radius <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$1-\epsilon $ </tex-math></inline-formula> and rate less than <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\frac {\epsilon }{2-\epsilon }$ </tex-math></inline-formula> , with constant list size. We deduce our results by extending and strengthening a recent result of Ferber, Kwan, and Sauermann on puncturing codes with large minimum distance and by utilizing the underlying code’s linearity.

Efficient multiplication of a vector by a matrix MDS

Abstract— An algorithm is proposed for the efficient multiplication of a vector by an n x n MDS matrix defined on Fq or by its inverse. The algorithm is based on the multiplication of two polynomials modulo a polynomial of degree n Reed-Solomon code generator and has complexity O(nlog2log2(nlog2q)). The algorithm only needs to store n values of the Fq field for the multiplication of a vector by an n x n MDS matrix and 2n values for the multiplication of the vector by the inverse matrix.

Resource reduction in multiplexed high-dimensional quantum Reed-Solomon codes

Quantum communication technologies will play an important role in quantum information processing in the near future as we network devices together. However, their implementation is still a challenging task due to both loss and gate errors. Quantum error correction codes are one important technique to address this issue. In particular, the Quantum Reed-Solomon codes are known to be quite efficient for quantum communication tasks. The high degree of physical resources required, however, makes such a code difficult to use in practice. A recent technique called quantum multiplexing has been shown to reduce resources by using multiple degrees of freedom of a photon. In this work, we propose a method to decompose multi-controlled gates using fewer $\rm{CX}$ gates via this quantum multiplexing technique. We show that our method can significantly reduce the required number of $\rm{CX}$ gates needed in the encoding circuits for the quantum Reed-Solomon code. Our approach is also applicable to many other quantum error correction codes and quantum algorithms, including Grovers and quantum walks.

Reversible extended secret image sharing with ability to correct errors based on Chinese remainder theorem

The reversible extended secret image sharing (RESIS) scheme can safely segment the secret image into a shadow image and embed it into the cover image, while ensuring that both the secret image and the cover image are completely restored. The existing schemes do not consider the attack on the information transmission channel, and often cannot correctly recover the secret image when attacked. In view of this, this paper fully considers the active attack on the information channel, and then proposes a RESIS scheme with error correction capability. In this paper, the Reed-Solomon code is used to detect modification attacks and correct errors to a certain extent. Additionally, the lossless recovery effect of both the secret image and the cover image is accomplished in conjunction with secret sharing scheme based on the Chinese remainder theorem. According to experimental findings, this method can resist certain active attacks.

Estimation for the number of MDS Matrices, Recursive MDS Matrices and Symmetric Recursive MDS Matrices from the Reed-Solomon Codes

The diffusion layer of the SPN block ciphers is usually built on the basis of the MDS (Maximum Distance Separable) matrices which is the matrix of the maximum distance separable code (MDS code). MDS codes have long been studied in error correcting code theory and have applications not only in coding theory but also in the design of block ciphers and hash functions. Thanks to that important role, there have been many studies on methods of building MDS matrices. In particular, the recursive MDS matrices and the symmetric recursive MDS matrices have particularly important applications because they are very efficient for execution. In this paper, we will give an estimate of the number of MDS matrices, recursive MDS matrices and symmetric recursive MDS matrices built from Reed-Solomon codes. This result is meaningful in determining the efficiency from this method of building matrices based on the Reed-Solomon codes. From there, this method can be applied to find out many MDS matrices, secure and efficient symmetric recursive MDS matrices for execution to apply in current block ciphers. Furthermore, recursive MDS matrices can be efficiently implemented using Linear Feedback Shift Registers (LFSR), making them well suited for lightweight cryptographic algorithms, so suitable for limited resources application.

Performance Analysis of Concatenated Coding to Increase the Endurance of Multilevel NAND Flash Memory

The increasing storage density of modern NAND flash memory chips, achieved both due to scaling down the cell size, and due to the increasing number of used cell states, leads to a decrease in data storage reliability, namely, error probability, endurance (number of P/E cycling) and retention time. Error correction codes are often used to improve the reliability of data storage in multilevel flash memory. The effectiveness of using error correction codes is largely determined by the model accuracy that exhibits the basic processes associated with writing and reading data. The paper describes the main sources of disturbances for a flash cell that affect the threshold voltage of the cell in NAND flash memory, and represents an explicit form of the threshold voltage distribution. As an approximation of the obtained threshold voltage distribution, a Normal-Laplace mixture model was shown to be a good fit in multilevel flash memories for a large number of rewriting cycles. For this model, a performance analysis of the concatenated coding scheme with an outer Reed-Solomon code and an inner multilevel code consisting of binary component codes is carried out. The performed analysis makes it possible to obtain tradeoffs between the error probability, storage density, and the number of P/E cycling. The resulting tradeoffs show that the considered concatenated coding schemes allow, due to a very slight decrease in the storage density, to increase the number of P/E cycling up to 2–2.5 times than their nominal endurance specification while maintaining the required value of the bit error probability.

Deep Learning Algorithms for BCH Decoding in Satellite Communication

Deep learning is widely used in various fields due to the advancement of algorithms, the enrichment of high-efficiency databases, and the increase in computing power. Especially in the satellite communication, the learning and parallel computing capabilities of neural networks make them ideal for decoding. Many researchers have recently applied deep learning neural networks to decode high-density parity check (HDPC) codes (such as BCH and RS code), improving the decoder’s performance. This review aims to provide general insights on applying neural network decoders to satellite communications. Due to the neural network’s learning ability, the neural network-based decoder can be trained to change the weights, thereby reducing the influence of non-white noise in satellite communications, such as the influence between the satellite and the terrestrial network and the mutual interference within the satellites. To compensate for non-white noise, shortest circles in Tanner graph and unreliable information, a decoder system model for satellite communication constructed by three neural networks is presented.

Comparison between dicode PPM and offset PPM enhancement with different error correction codes

AbstractOffset pulse position modulation (OPPM) and Dicode pulse position modulation (DiPPM) have been introduced as new attractive modulation techniques, however, they are suffering from erasure, wrong slot, and false alarm errors. in this paper, two types of error correction (EC) codes, Low‐Density Parity Check (LDPC) code and Reed Solomon (RS) code, where paired the OPPM and DiPPM systems to reduce bugs, analyze the best EC parameters, and choose the superlative EC system. In other words, the performance of OPPM engaging LDPC codes and DiPPM employing LDPC codes will be compared against the OPPM engaging RS codes and DiPPM employing RS codes. To systematically carry out this comparison, numbers of photons, transmission efficiency, and the code rate are computed. The evaluation has stated that the data transmission rate at the start has indicated that OPPM with LDPC requires only 1.2 × 103 photons/pulse compared to the necessity of 1.821 × 103 photons/pulse when the DiPPM used LDPC. Accordingly, the transmission efficiency has been increased due to a reduction of number of photons. Also, the coded OPPM with LDPC codes is better than the coded DiPPM with LDPC codes when operating at a code rate that is approximately 0.7.

Performance Evaluation of Burst Error Correction by LDPC Coding and Iterative Decoding System in Magnetic Tape Drive

The tolerance for burst error of interleaved low-density parity-check (LDPC) code is evaluated with the cross-interleaved Reed–Solomon code (CIRC) in the magnetic tape drive using a BaFe magnetic tape. We evaluate the interleaved LDPC coding and iterative decoding system with burst error detector (LDPC-BD) system by the required signal-to-noise ratio (SNR) to achieve “No errors” for the burst-like deterioration such as media defects to the CIRC with erasure error correction (CIRC-EC) system by computer simulation. The results clarify that the LDPC-BD system tolerates the longer burst errors than the CIRC-EC system and the LDPC-BD system achieves “No errors” with the 3.0–4.5 dB lower SNR than the CIRC-EC system.

Orbit Structure of Grassmannian G2,m and a Decoder for Grassmann Code C(2, m).

In this article, we consider decoding Grassmann codes, linear codes associated to the Grassmannian and its embedding in a projective space. We look at the orbit structure of Grassmannian arising from the multiplicative group in . We project the corresponding Grassmann code onto these orbits to obtain a subcode of a -ary Reed-Solomon code. We prove that some of these projections contain an information set of the parent Grassmann code. By improving the decoding capacity of Peterson's decoding algorithm for the projected subcodes, we prove that one can correct up to errors for Grassmann code, where is the minimum distance of Grassmann code.

Some quantum MDS codes from GRS codes

ABSTRACT The construction of quantum maximum-distance-separable (MDS) codes has become one of the major goals in quantum coding theory. In this paper, we construct several classes of Hermitian self-orthogonal generalized Reed–Solomon codes. Based on these classical MDS codes, we obtain several new classes of quantum MDS codes with large minimum distance. It turns out that these constructed quantum MDS codes have fewer constraints on the selection of code length, and in some cases, have larger minimum distance than previous literature. Notably, some of the distance parameters of our codes are greater than ( q / 2 ) + 1 .