Erasure decoding of five times extended Reed-Solomon codes

Applicable Algebra in Engineering, Communication and Computing, 1994

The subject of decoding Reed-Solomon codes is considered, By reformulating the Berlekamp and Welch key equation and introducing new versions of this key equation, two new decoding algorithms for Reed-Solomon codes will be presented. The two new decoding algorithms are significant for three reasons. Firstly the new equations and algorithms represent a novel approach to the extensively researched problem of decoding Reed-Solomon codes. Secondly the algorithms have algorithmic and implementation complexity comparable to existing decoding algorithms, and as such present a viable solution for decoding Reed-Solomon codes. Thirdly the new ideas presented suggest a direction for future research. The first algorithm uses the extended Euclidean algorithm an~t is very efficient for a systolic VLSI implementation. The second decoding algorithm presented is similar in nature to the original decoding algorithm of Peterson except that the syndromes do not need to be computed and the remainders are used directly. It has a regular structure and will be efficient for implementation only for correcting a small number of errors. A systolic design for computing the Lagrange interpolation of a polynomial, which is needed for the first decoding algorithm, is also presented.

IEEE Letters of the Computer Society, 2019

In this letter it is proved that five times extended Reed-Solomon codes contain an infinite subset of almost MDS codes with parameters: ½ðq À 1Þ þ 5; ðq À 1Þ; 5 GF ð2 m Þ which are defined over a finite field GF ð2 m Þ where m ! 3 is a positive odd integer. The first three of these codes reach an upper bound for linear block code distance for the corresponding codeword lengths and number of information symbols in codewords in existing tables for optimal code parameters [1]. The relevant parts of weight spectra for the first four codes confirm the code parameters.

2007

The research that led to this thesis was inspired by Sudan's breakthrough that demonstrated that Reed-Solomon codes can correct more errors than previously thought. This breakthrough can render the current state-of-the-art Reed-Solomon decoders obsolete. Much of the importance of Reed-Solomon codes stems from their ubiquity and utility. This thesis takes a few steps toward a deeper understanding of Reed-Solomon codes as well as toward the design of efficient algorithms for decoding them. After studying the binary images of Reed-Solomon codes, we proceeded to analyze their performance under optimum decoding. Moreover, we investigated the performance of Reed-Solomon codes in network scenarios when the code is shared by many users or applications. We proved that Reed-Solomon codes have many more desirable properties. Algebraic soft decoding of Reed-Solomon codes is a class of algorithms that was stirred by Sudan's breakthrough. We developed a mathematical model for algebraic so...

2004

The standard Berlekamp-Massey iterative algorithm, although the most commonly used decoding method for Reed-Solomon (RS) codes, is computationally complex. A novel decoding technique that combines error trapping decoding and the Berlekamp-Massey algorithm is proposed in this article. It is demonstrated that this decoder maintains the simplicity of the error trapping decoder for the most part. Moreover, it is shown that whenever it is necessary to apply the maximum error correcting capability of the code, only a shortened Berlekamp-Massey algorithm is utilized. Substantial improvement in the decoder throughput relative to that of Berlekamp-Massey decoding is reported. Software simulation is used to verify the theoretical performance analysis.

2003

In this paper, we introduce a new family of linear block codes, which we refer to as Partial Reed Solomon (PRS) Codes. These codes are specifically designed and optimized for real-time multimedia communication over packet-based erasure channels. Based on the constraints and flexibilities of real time applications, we define a performance measure, message throughput (m τ), which is suitable for these applications. This measure differentiates the notion of optimum codes for the target multimedia applications as compared to performance measures that are used for non-realtime data. Based on the proposed measure, we combine the advantages of lowering the density of a code for near capacity performance with the high decoding efficiency of Reed Solomon (RS) codes, in order to design optimum PRS codes. Then, we demonstrate, through an example of a Binary Erasure Channel (BEC), that at near-capacity coding rates, appropriate design of a PRS code can outperform an RS-code. We extend this analysis and optimization for a general BEC over a wide range of channel conditions. Moreover, as compared with RS codes, the proposed PRS codes provide a significantly improved graceful degradation when the number of losses exceeds the number of parity symbols within the code block. This is a highly desirable feature for realtime multimedia applications. I.

INTERNATIONAL JOURNAL OF ENGINEERING DEVELOPMENT AND RESEARCH (IJEDR) (ISSN:2321-9939), 2014

Reed-Solomon codes are very useful and effective in burst error in noisy environment. In decoding process for 1 error or 2 errors create easily with using procedure of Peterson-Gorenstein –Zierler algorithm. If decoding process for 3 or more errors, these errors can be solved with key equation of a new algorithm named Berlekamp-massey algorithm. In this paper, wide discussion of procedures of Peterson-Gorenstein –Zierler algorithm and Berlekamp-Massey algorithm and show the advantages of modified version of Berlekam-Massey algorithm with its steps.

2008 IEEE International Symposium on Information Theory, 2008

Generalized Concatenated codes are a code construction consisting of a number of outer codes whose code symbols are protected by an inner code. As outer codes, we assume the most frequently used Reed-Solomon codes; as inner code, we assume some linear block code which can be decoded up to half its minimum distance. Decoding up to half the minimum distance of Generalized Concatenated codes is classically achieved by the Blokh-Zyablov-Dumer algorithm, which iteratively decodes by first using the inner decoder to get an estimate of the outer code words and then using an outer error/erasure decoder with a varying number of erasures determined by a set of precalculated thresholds. In this paper, a modified version of the Blokh-Zyablov-Dumer algorithm is proposed, which exploits the fact that a number of outer Reed-Solomon codes with average minimum distanced can be grouped into one single Interleaved Reed-Solomon code which can be decoded beyondd/2. This allows to skip a number of decoding iterations on the one hand and to reduce the complexity of each decoding iteration significantly -while maintaining the decoding performance -on the other.

IEEE Transactions on Communications, 2003

This paper presents a computationally efficient hybrid reliability-based decoding algorithm for Reed-Solomon (RS) codes. This hybrid decoding algorithm consists of two major components, a reencoding process and a successive erasure-and-error decoding process for both bit and symbol levels. The reencoding process is to generate a sequence of candidate codewords based on the information provided by the codeword decoded by an algebraic decoder and a set of test error patterns. Two criteria are used for testing in the decoding process to reduce the decoding computational complexity. The first criterion is devised to reduce the number of reencoding operations by eliminating the unlikely error patterns. The second criterion is to test the optimality of a generated candidate codeword. Numerical results show that the proposed decoding algorithm can achieve either a near-optimum error performance or an asymptotically optimum error performance.

Designs, Codes and Cryptography, 2012

A new lower bound on the minimum distance of qary cyclic codes is proposed. This bound improves upon the Bose-Chaudhuri-Hocquenghem (BCH) bound and, for some codes, upon the Hartmann-Tzeng (HT) bound. Several Boston bounds are special cases of our bound. For some classes of codes the bound on the minimum distance is refined. Furthermore, a quadratic-time decoding algorithm up to this new bound is developed. The determination of the error locations is based on the Euclidean Algorithm and a modified Chien search. The error evaluation is done by solving a generalization of Forney's formula.

IEEE Transactions on Communications, 2000

A simplified parallel step-by-step decoding algorithm is proposed for decoding Reed-Solomon (RS) codes. It uses new method to calculate the determinants of the temporarily changed syndrome matrices, based on the property of these matrices determined in this paper. By using the proposed method, the calculations of the determinants of the temporarily changed syndrome matrices become much simpler and thus the computational complexity of the step-by-step decoding algorithm is significantly reduced.

2014 IEEE 55th Annual Symposium on Foundations of Computer Science, 2014

In this paper, we present a new basis of polynomial over finite fields of characteristic two and then apply it to the encoding/decoding of Reed-Solomon erasure codes. The proposed polynomial basis allows that h-point polynomial evaluation can be computed in O(h log 2 (h)) finite field operations with small leading constant. As compared with the canonical polynomial basis, the proposed basis improves the arithmetic complexity of addition, multiplication, and the determination of polynomial degree from O(h log 2 (h) log 2 log 2 (h)) to O(h log 2 (h)). Based on this basis, we then develop the encoding and erasure decoding algorithms for the (n = 2 r , k) Reed-Solomon codes. Thanks to the efficiency of transform based on the polynomial basis, the encoding can be completed in O(n log 2 (k)) finite field operations, and the erasure decoding in O(n log 2 (n)) finite field operations. To the best of our knowledge, this is the first approach supporting Reed-Solomon erasure codes over characteristic-2 finite fields while achieving a complexity of O(n log 2 (n)), in both additive and multiplicative complexities. As the complexity leading factor is small, the algorithms are advantageous in practical applications.

Journal of Complexity, 1997

We present a randomized algorithm which takes as input n distinct points f(x ; y )g from F 2 F (where F is a field) and integer parameters t and d and returns a list of all univariate polynomials f over F in the variable x of degree at most d which agree with the given set of points in at least t places (i.e., y = f (x ) for at least t values of i), provided t = ( p nd). The running time is bounded by a polynomial in n. This immediately provides a maximum likelihood decoding algorithm for Reed Solomon Codes, which works in a setting with a larger number of errors than any previously known algorithm. To the best of our knowledge, this is the first efficient (i.e., polynomial time bounded) algorithm which provides error recovery capability beyond the error-correction bound of a code for any efficient (i.e., constant or even polynomial rate) code.

A recently presented algorithm for joint decoding of interleaved Reed-Solomon (IRS) codes based on the extended Euclidean algorithm is recalled and it will be shown that with a simple extension a list decoding algorithm is obtained. It is also shown how decoding works in case when the error weight is not the same for all codewords.

IEEE Transactions on Communications, 2007

A simplified parallel step-by-step decoding algorithm is proposed for decoding Reed-Solomon (RS) codes. It uses new method to calculate the determinants of the temporarily changed syndrome matrices, based on the property of these matrices determined in this paper. By using the proposed method, the calculations of the determinants of the temporarily changed syndrome matrices become much simpler and thus the computational complexity of the step-by-step decoding algorithm is significantly reduced.