Simplified procedure for decoding nonsystematic reed-solomon codes over gf(2m) using euclid's algorithm and the fast fourier transform

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.

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.

Reed Solomon codes are being widely used for error-correcting methods and in construction of fault tolerant systems such as RAID. Recent studies[2],[7],[8] revealed that these codes have a potential application in cryptography. The hardness in decoding Reed Solomon codes can be exposed to construct new cryptographic applications. It is in this scenario that we talk about the suitability of Reed Solomon codes for Digital Fingerprinting in particular and cryptography in general. We derive results for bounds on Fingerprint length, design tracing method for Boneh.s broadcast encryption system[2]; and discuss collusion attacks on fingerprint. Pirate strategies and performance issues are also presented.KeywordsEncryption SchemeSecret MessageReed Solomon CodeDetectable MarkElliptic Curve Discrete Logarithm ProblemThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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.

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.

There has been renewed interest in decoding Reed-Solomon (RS) codes without using syndromes recently. In this paper, we investigate the complexity of syndromeless decoding, and compare it to that of syndrome-based decoding. Aiming to provide guidelines to practical applications, our complexity analysis focuses on RS codes over characteristic-2 fields, for which some multiplicative FFT techniques are not applicable. Due to moderate block lengths of RS codes in practice, our analysis is complete, without big O notation. In addition to fast implementation using additive FFT techniques, we also consider direct implementation, which is still relevant for RS codes with moderate lengths. For high-rate RS codes, when compared to syndrome-based decoding algorithms, not only syndromeless decoding algorithms require more field operations regardless of implementation, but also decoder architectures based on their direct implementations have higher hardware costs and lower throughput. We also derive tighter bounds on the complexities of fast polynomial multiplications based on Cantor's approach and the fast extended Euclidean algorithm.

Non-binary multithreshold decoding (qMTD) for q-ary self-orthogonal codes (qSOC) is considered. The SER performance of qMTD is shown to be close to the results provided by optimum total search methods, which are not realizable for non-binary codes in general. qMTD decoders are compared with different decoders for Reed-Solomon codes. The performance provided with qMTD in some cases is unattainable with classical decoders for arbitrary long Reed-Solomon codes. The result of concatenation of qSOC with simple to decode outer codes is described. Method of improving of qMTD decoder's performance for qSOC is offered. Some simulated results obtained by using these two decoding techniques (the base and modified ways) are presented as well. Comparison of the results showed that the change in the threshold element's algorithm can significantly improve speed of qMTD work. It's shown that for a larger gain this modification qMTD should be used after conventional decoding iterations.

A parallel linear generator of multi-valued pseudorandom sequences, which operates under conditions of generating hardware errors caused by destructive adversary actions is proposed. The main types of modification of the pseudorandom sequence in case of adversary attack are considered. A distinctive feature of the iterative process of ensuring the reliability of computational operations is the of computational operations by representing a system of generating recurring logical formulas as a system of many-valued logic algebra functions. The subsequent realization of multivalued logic algebra functions by means of arithmetic polynomials allowed us to parallelize the process of generating multivalued pseudorandom sequences and level out the existing complexity (specificity) of cryptographic transformations of logical data types which limit the use of redundant coding methods. As a result, a solution that allows to apply redundant modular codes to control the accuracy of the computational operations performed by the nodes of pseudorandom sequence generation is proposed. Moreover, unlike the known solutions, the proposed method provides obtaining fragments of a pseudorandom sequence on the basis of one recursive arithmetic formula with parallel calculation errors control. The use of modular forms made it possible to transfer computations from the rational numbers field arithmetic to integer arithmetic of a simple field. Among the existing variety of codes correcting errors (maximally spaced codes), a special place is occupied by multivalued Reed-Solomon codes. Reed-Solomon codes usage in the formation of pseudorandom sequences allows the formation of code-like structures that monitor and ensure the reliability of computational operations. The calculated probability of failure-free operation of the parallel linear generator of multivalued pseudorandom sequences with an error control function based on the principle of functioning — sliding redundancy is obtained. The achieved results can find wide application at realization of perspective high-efficiency cryptographic information protection facility.

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

Polar codes are the first provable capacity-achieving channel codes. Despite the splendid performance of long Polar codes, short Polar codes have relatively poor performance compared with other modern channel coding schemes (e.g., Turbo codes and LDPC). In this paper, we explore some practical methods to improve the performance of Polar codes with short to moderate codeword lengths. First, we use Reed Solomon (RS) codes as outer codes. With a specific interleaving strategy, we can concatenate multiple RS codes with one frame of Polar codes. Combining a strategy of allocating unequal RS code rates with the concatenation, different levels of protection are assigned based on the error pattern of successive cancellation list (SCL) decoders. Thus, the finite length performance will certainly be enhanced for this encoding scheme. Meanwhile, the memory size that the original SCL decoding procedure requires is reduced, and the increment of overall decoding complexity is small. Additionally, we propose an intra-frame interleaver to further enhance the performance by dispersing errors. Finally, we designed a list decoding scheme for the proposed multi-RS concatenated Polar codes. Depending on the soft information generated by an SCL decoder, we calculated the reliability of each RS symbol and conducted soft RS decoding. So, the overall performance was enhanced under this joint decoding strategy. Simulation results indicate that the bit error rate (BER) performance of short Polar codes can be well improved.

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.

Previously, the authors proposed an inverse-free Berlekamp-Massey (1968, 1969) algorithm to simplify the Reed-Solomon (RS) codes. This modified RS decoding method is the best known technique for finding the error locator polynomial. The inverse-free method is generalized to find both errors and erasures. The basic idea of the new procedure is the replacement of the initial condition of the BM algorithm by the Forney (1965) syndromes. With this improved technique, the complexity of time-domain RS decoders for correcting both errors and erasures is reduced substantially from previous approaches.

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.

We construct an explicit family of linear rank-metric codes over any field F that enables efficient list decoding up to a fraction rho of errors in the rank metric with a rate of 1-rho-eps, for any desired rho in (0,1) and eps > 0. Previously, a Monte Carlo construction of such codes was known, but this is in fact the first explicit construction of positive rate rank-metric codes for list decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an F-subspace that evades certain structured subspaces over an extension field of F. These structured spaces arise from the linear-algebraic list decoder for Gabidulin codes due to Guruswami and Xing (STOC'13). Our construction is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which are a collection of subspaces, every pair of which have low-dimensional intersection, and which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order that are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list decoding RS codes reduces to list decoding such folded RS codes. However, as we only list decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list decoding RS codes.

The telecommunication systems development based on single-mode and multimode optical fibers is aimed at the rapidly increasing demand for bandwidth not only transport networks, but also intra-site data centers networks, cloud computing, and web servers. The solution to this problem is provided by the signal processing technologies complexity. In this case, the requirements for the receiver complexity can be reduced by using technologies to combat signal distortion due to equalizers together with FEC. There was known that the use of FEC based on permutation decoding in relation to the RS codes provides a significant gain in the data processing speed by the receiver due to the use of a cognitive procedure for searching the generating matrix of the equivalent code. The paper shows that the use of this technology can be successful in solving another important task related to minimizing the timing attacks consequences on the protection confidential information system. This is due to the active development of quantum computers and the problems of post-quantum cryptography that arose on this basis. The essence of leveling the timing attacks risks is to equalize the time spent on processing vectors of redundant BCH or RS codes. The use of permutation decoding fully matches these requirements by replacing the algebraic system of detection and errors correction in the data on the procedure of lexicographic search for the desired result. It is shown that the data processing time when using permutation decoding in conjunction with the cognitive search for the decoding result varies slightly.