Reed Solomon Codes for Digital Fingerprinting

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.

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.

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

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.

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 fast decoding algorithm for double error correcting Reed Solomon (RS) codes is presented. A new approach to the RS code decoding is introduced to eliminate the Chien search, to reduce substantially the root search time and hence to increase the decoding speed. It is interesting that the error locating time almost constant as the number of bits per symbol is increased. The algorithm finds suitable for microprocessor-based as well as VLSI implementations. Various RS codes are considered. Complexity, memory space and decoding time are all minimized as compared to the normal Chien search approach.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

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.

MDS matrices are from the MDS codes in coding theory that are being used widely in cryptographic applications. Recursive MDS matrix is an matrix that is a power of some simple companion matrix. These matrices are so convenient for execution especially for hardware implementation using LFSRs. Therefore, these matrices have attracted the interest of many scientists. In this paper, we give a way to directly build 8 × 8 self-reciprocal recursive MDS matrices efficient for execution over the field GF(q), (q = pr, pis a prime number) using the Reed-Solomon codes. These matrices are significant in practice because they have the potential to be used in lightweight cryptographic algorithms.

The error-correcting pair for TGRS codes

Broadcast encryption is a cryptographic primitive that allows a sender to securely transmit a message to a set of receivers. The most influential broadcast encryption system was proposed in 2001 by Naor, Naor, Lotspiech, based on binary trees and the Subset Difference (SD) method. In 2006, Jang, Nyang, and Song suggested a new broadcast encryption system that can reduce transmission rate by 50% compared to the SD method, by introducing the so-called '2-SD' method. Their result was later given the registration of a patent in Korea (registration number: 100879083). Unfortunately, however, this paper shows that Jang et. al.'s broadcast encryption system is not secure against collusion attacks that are considered as being the basic security requirement in designing broadcast encryption.

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.

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.

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.