An approach to double error correcting Reed-Solomon decoding without Chien search

Reed Solomon (RS) codes are used in a variety of (wireless) communication systems. Although commonly implemented in dedicated hardware, this paper explores the mapping of high-throughput RS decoding on vector DSPs. The four modules of such a decoder, viz. syndrome computation, key equation solver, Chien search, and Forney pose different vectorization challenges. Their vectorizations are explained in detail, including optimizations specific for embedded vector processor (EVP). For RS (255,239), this solution is benchmarked vs published implementations, and scalability up to vector size 64 is explored. The best and the worst case throughput of our implementation is 8 times and 2 times higher respectively than other architectures.

In this paper, a blind reconstruction method of Bose-Chaudhuri-Hocquenghem (BCH) and Reed-Solomon (RS) codes is proposed, which aims to reconstruct the generator polynomials of BCH and RS codes from the intercepted noisy bitstream in non-cooperative environments. The previous blind reconstruction methods try to select correct (or noiseless) codewords or extract code information from the received bitstream. However, they need to have sufficiently correct codewords for successful reconstruction. Therefore, the existing methods are highly dependent on the channel conditions because a sufficient number of correct codewords should be available for achieving good reconstruction performance. To overcome such high dependency on channel conditions, the proposed blind reconstruction method performs error correction of single-error codewords through the proposed single-error correction method (SCM). Then, both the correct codewords and the corrected single-error codewords are utilized for blind reconstruction. Specifically, the correct codewords are selected based on the algebraic property of BCH and RS codes that codeword polynomials have consecutive roots. At the same time, the codewords presumably with a single error are selected and corrected based on the fact that the most of BCH and RS codeword polynomials with either a single error or double errors do not have consecutive roots. Based on this fact, SCM is devise, which selects and corrects the codewords with a single error with high probability. Finally, it is verified that the proposed blind reconstruction method shows much better reconstruction performance by using only one-tenth of the received codewords compared to the existing best methods even under harsh channel conditions.

Maximum distance separable (MDS) codes are those codes whose minimum distance attains the Singleton bound for a given length and given dimension. It is a well-known fact that the only binary linear block codes that are MDS are the simple parity codes and the repetition codes. Known MDS codes (the Reed-Solomon codes, for instance) are defined over larger alphabets. This approach requires that (i) the encoding and decoding procedures are performed as operations over a finite field and (ii) an update in a single information bit (e.g. in storage applications) requires an update in the redundancy symbols and usually affects a number of bits in each redundancy symbol. Thus, the optimal redundancy is achieved at the expense of additional complexity in the encoding/decoding procedures as well as in the number of redundancy bits affected by an update in the information. We construct MDS codes that have the following two properties: (i) the redundancy bits are computed by simple XOR operations; and (ii) an update in an information bit affects a minimal number of redundancy bits. >

This paper proposes a generator polynomial model-based eye diagram estimation method for the error-correction codes (ECCs), which are the Bose–Chaudhuri–Hocquenghem (BCH) code and the Reed–Solomon (RS) code. The ECCs compensate for the channel loss at the bit level, so most of the systems include the ECC block. The BCH and RS codes are the generator polynomial-based ECCs; this paper proposes the generator polynomial model-based method. The model provides the accurate probability density function of the code words for the BCH and RS code. Thus, the proposed method is capable of achieving the signal integrity analysis for the system level. For verification, we design the BCH and RS encoders. The estimated eye diagrams are nearly the same as that of the transient simulation for the BCH and RS codes. Furthermore, the voltage bathtub curves are the same up to 10−8. For a smaller bit-error rate value, the proposed method only provides the bathtub curve. This paper successfully verifies the eye diagram estimation method for the BCH and RS codes.

The repair problem in distributed storage addresses recovery of the data encoded using an erasure code, for instance, a Reed-Solomon (RS) code. We consider the problem of repairing a single node or multiple nodes in RS-coded storage systems using the smallest possible amount of inter-nodal communication. According to the cut-set bound, communication cost of repairing h ≥ 1 failed nodes for an (n, k = n - r) maximum distance separable (MDS) code using d helper nodes is at least dhl/(d + h - k), where l is the size of the node. Guruswami and Wootters (2016) initiated the study of efficient repair of RS codes, showing that they can be repaired using a smaller bandwidth than under the trivial approach. At the same time, their work as well as follow-up papers stopped short of constructing RS codes (or any scalar MDS codes) that meet the cut-set bound with equality. In this paper, we construct the families of RS codes that achieve the cut-set bound for repair of one or several nodes. In the single-node case, we present the RS codes of length n over the field F(q <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">l</sup> ), l = exp((1 + o(1))n logn) that meet the cut-set bound. We also prove an almost matching lower bound on l, showing that super-exponential scaling is both necessary and sufficient for scalar MDS codes to achieve the cut-set bound using linear repair schemes. For the case of multiple nodes, we construct a family of RS codes that achieve the cut-set bound universally for the repair of any h = 1,2, . . ., r failed nodes from any subset of d helper nodes, k ≤ d ≤ n - h. For a fixed number of parities r, the node size of the constructed codes is close to the smallest possible node size for codes with such properties.

Decoding algorithms for Reed--Solomon (RS) codes are of great interest for both practical and theoretical reasons. In this paper, an efficient algorithm, called the modular approach (MA), is devised for solving the Welch--Berlekamp (WB) key equation. By taking the MA as the key equation solver, we propose a new decoding algorithm for systematic RS codes. For $(n,k)$ RS codes, where $n$ is the code length and $k$ is the code dimension, the proposed decoding algorithm has both the best asymptotic computational complexity $O(n\log(n-k) + (n-k)\log^2(n-k))$ and the smallest constant factor achieved to date. By comparing the number of field operations required, we show that when decoding practical RS codes, the new algorithm is significantly superior to the existing methods in terms of computational complexity. When decoding the $(4096, 3584)$ RS code defined over $\mathbb{F}_{2^{12}}$, the new algorithm is 10 times faster than a conventional syndrome-based method. Furthermore, the new algorithm has a regular architecture and is thus suitable for hardware implementation.

Fading in mobile satellite communications severely degrades the performance of data transmission. The emphasis in this paper is on the performance of uncoded and coded FCMA (frequency comb multiple access) in the fading environment. The channel is modeled with nonfrequency selective Rician and Rayleigh fading using noncoherent demodulation with Reed-Solomon (RS) codes and hard decision decoding. Berlekamp-Massey decoding of RS codes is used to compensate for the fading. The bit error rate (BER) after decoding is calculated for specific codes and for different values of Rician channel parameters. The results of this paper are obtained by simulation techniques. They show that substantial coding gains are obtained compared to the uncoded reference system. They are also useful as reference for validating the results of simulation studies.

Many images you see on the Internet today have undergone compression for various reasons. Image compression can benefit users by having pictures load faster and webpages use up less space on a Web host. Image compression does not reduce the physical size of an image but instead compresses the data that makes up the image into a smaller size. In case of image transmission the noise will decrease the quality of recivide image which obliges us to use channel coding techniques to protect our data against the channel noise. The Reed-Solomon code is one of the most popular channel coding techniques used to correct errors in many systems ((Wireless or mobile communications, Satellite communications, Digital television / DVB,High-speed modems such as ADSL, xDSL, etc.). Since there is lot of possibilities to select the input parameters of RS code this will make us concerned about the optimum input that can protect our data with minimum number of redundant bits. In this paper we are going to use the genetic algorithm to optimize in the selction of input parameters of RS code acording to the channel conditions wich reduce the number of bits needed to protect our data with hight quality of received image.

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.

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.

In this chapter, we will give information about Reed-Solomon codes. These codes fall into the category of nonbinary cyclic codes. The generator polynomials of Reed-Solomon codes are constructed using the minimal polynomials of the extended finite fields. Reed-Solomon codes are effective for burst errors and they are used for erasure decoding. Reed-Solomon codes are used in some electronic devices such as CDs, DVDs, and Blu-ray, and they are also employed in communication technologies such as DSL, WiMAX, or RAID 6. Reed-Solomon codes are invented in 1960, and they are seen as nonbinary BCH codes. In this chapter, we first explain the construction of the generator polynomials of the Reed-Solomon codes using the minimal polynomials of the extended finite fields. Next, we provide information about the syndrome decoding of Reed-Solomon codes using error evaluator polynomial. In sequel, Berlekamp algorithm which is a low-complexity syndrome decoding algorithm used for the decoding of Reed-Solomon codes is explained.

Cloud data centers should be flexible and available to the data forever. The replication method is used to achieve high availability and durability of cloud data center, if there is any failure to recover the messages from the cloud databases. The concern of this replication technology is that, the replica size is equal to the size of the original data object. When Error Detection Schemes were used, there is a reduction in the number of cloud distributed storage systems. The scope of this paper is to store the data efficiently in cloud data centers unlike the previous schemes which used erasure codes such as Reed Solomon codes only with a view to store data in datacenters. This paper proposes to encrypt the message using DES and to encode the message using Reed Solomon code before storing the message. Storing time is convincingly good in Reed Solomon code when compared with tornado code.

In Chapter 6, we studied Reed–Solomon codes, codes whose codewords are the evaluation of polynomials in one variable of degree at most k − 1 at the elements of \({\mathbb F}_q \cup \{\infty \}\). Reed–Solomon codes are short length codes, where the length n is bounded by q + 1, and only useful when we take the field to be large. The alternant codes which we constructed from generalised Reed–Solomon codes in Chapter 7 allowed us to construct codes over small fields and we put this to good use. In this chapter we will consider another generalisation of Reed–Solomon codes, codes whose codewords are the evaluation of polynomials in many variables. This again allows us to construct linear codes over small fields and we will restrict our attention, for the most part, to binary linear codes. It will turn out that these codes are not asymptotically good. Nevertheless, they are an important class of codes which are widely implemented due to the availability of fast decoding algorithms. One example of such a decoding algorithm is the majority-logic decoding algorithm that we will study here. We will then go on and construct Kerdock codes which are certain subcodes of the second-order Reed–Muller codes. These codes can give examples of non-linear codes with parameters for which no linear code exists.

We consider (n, k , l) MDS codes of length n, dimension k, and subpacketization l over a finite field <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> . A codeword of such a code consists of n column-vectors of length l over <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">F</i> , with the property that any k of them suffice to recover the entire codeword. Each of these n vectors may be stored on a separate node in a network. If one of the n nodes fails, we can recover its content by downloading symbols from the surviving nodes, and the total number of symbols downloaded in the worst case is called the repair bandwidth of the code. By the cut-set bound, the repair bandwidth of an (n, k , l) MDS code is at least l(n-1)/(n- k). There are several constructions of MDS codes whose repair bandwidth meets or asymptotically meets the cut-set bound. For example, letting r=n-k denote the number of parities, Ye and Barg constructed (n, k, r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) Reed-Solomon codes that asymptotically meet the cut-set bound, Ye and Barg also constructed optimal-bandwidth and optimal-update (n, k, r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sup> ) MDS codes. Wang, Tamo, and Bruck constructed optimal-bandwidth (n, k, r <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> n/(r+1)</sup> ) MDS codes, and these codes have the smallest known subpacketization for optimal-bandwidth MDS codes. A key idea in all these constructions is to represent certain integers in base r. We show how this technique can be refined to improve the subpacketization of the two MDS code constructions by Ye and Barg, while achieving asymptotically optimal repair bandwidth. Specifically, when r=s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> for an integer s, we obtain an (n, k, s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m+n-1</sup> ) Reed-Solomon code and an optimal-update (n, k, s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m+ n-1</sup> ) MDS code, both having asymptotically optimal repair bandwidth. Thus for r = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> , for example, we achieve the subpacketization of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> m+n-1</sup> rather than 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>mn</i></sup> in the original constructions by Ye and Barg. When r is not an integral power, we can still obtain (n, k, s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> m+n-1</sup> ) Reed-Solomon codes and optimal-update ( n, k, s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m+ n-1</sup> ) MDS codes by choosing positive integers s and m such that s <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m </sup> ≤ r. In this case, however, the resulting codes have bandwidth that is near-optimal rather than asymptotically optimal. We also present an extension of this idea to reduce the subpacketization of the Wang-Tamo-Bruck construction while achieving a repair-by-transfer scheme with asymptotically optimal repair bandwidth. For example, for r = 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">m</sup> we achieve the subpacketization of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">k/r+ m-1</sup> , which significantly improves upon the subpacketization of 2 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>mn</i>/(r+1)</sup> in the Wang-Tamo-Bruck construction. Based on the foregoing examples, we believe our approach may be generally useful in reducing the subpacketization of MDS code constructions that utilize r-ary expansion.

Techniques are presented for computing upper and lower bounds on the number of errors that can be corrected by list decoders for general block codes and, specifically, for Reed--Solomon (RS) codes. The list decoder of Guruswami and Sudan implies such a lower bound (referred to here as the GS bound) for RS codes. It is shown that this lower bound, given by means of the code's length, the minimum Hamming distance, and the maximal allowed list size, in fact applies to all block codes. Ranges of code parameters are identified where the GS bound is tight for worst-case RS codes, in which case the list decoder of Guruswami and Sudan provably corrects the largest possible number of errors. On the other hand, ranges of parameters are provided for which the GS lower bound can be strictly improved. In some cases the improvement applies to all block codes with a given minimum Hamming distance, while in others it applies only to RS codes.