Decoding Of Reed-Solomon Codes Research Articles

Software optimization for fast encoding and decoding of Reed-Solomon codes

In this work, we propose a software library written in C for encoding and decoding Reed-Solomon codes. Library consists of one scalar CODEC and two vectorized codecs for x86 architecture. Vectorized codecs use the benefits of SSSE3 or AVX2 instruction sets. We compare the performance of our three codecs with the JPWL RS CODEC from the Open JPEG library. The performance comparison methodology is described, and it is based on the measuring of the encoding and decoding speed. The results demonstrate a 4.1x speed gain with the scalar CODEC and a 19.6x gain with the vectorized CODEC. Based on testing results and supported instruction sets, a dynamic selection of CODEC version is proposed.

On fast Fourier transform-based decoding of Reed-Solomon codes

Reed-Solomon (RS) codes is a popular class of codes that have been implemented in many practical systems. Recently, a fast approach to the error decoding of RS codes based on fast Fourier transform (FFT) was invented. In this work, we derive the key equation based on the Lagrange polynomial and then present erasure-and-error decoding of an (n; k) RS code. This decoding algorithm can simultaneously correct up to v errors and f erasures when 2v + f < n − k + 1. The decoding complexity is with only O(n log n + (n − k) log2(n − k)).

On Soft-Information-Based Error and Erasure Decoding of Reed–Solomon Codes in Burst Rayleigh Fading Channels

In this paper, two new decoding algorithms to decode Reed–Solomon codes during transmission over burst Rayleigh fading channels with additive white Gaussian noise (AWGN) are proposed. They only conduct error correction for coded symbols located in the pure AWGN region and conduct error and erasure correction for those symbols located in the burst fading region by treating those coded symbols that are very likely erroneous as erasures. The first algorithm does not need to know the fading locations in advance, while the second algorithm assumes that the fading locations are known. In addition, the performance of such two algorithms is studied when a pre-computed threshold is used to determine the erasures of the code. Simulation results show that our proposed algorithms not only significantly perform better than the classic Berlekamp–Messay algorithm with a comparable computational complexity but also achieve a better tradeoff between the performance and the computational complexity when compared with other existing algorithms. In particular, our algorithms exhibit excellent robustness for tested various code parameters and fading configurations. Furthermore, a more detailed mathematical analysis is also developed in this paper in order to estimate the performance of the new algorithms in the burst Rayleigh fading channels. We observe that the performance of the first algorithm can only be estimated relatively accurately when encountering burst deep-fading, whereas the performance prediction for the second algorithm is always in agreement with the simulation results for various fading cases.

Efficiently Decoding Reed–Muller Codes From Random Errors

Reed–Muller (RM) codes encode an $m$ -variate polynomial of degree at most $r$ by evaluating it on all points in $\{0,1\}^{m}$ . We denote this code by $RM(r,m)$ . The minimum distance of $RM(r,m)$ is $2^{m-r}$ and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this paper we give an efficient algorithm (in the block length $n=2^{m}$ ) for decoding random errors in RM codes far beyond the minimum distance. Specifically, for low-rate codes (of degree $r=o(\sqrt {m})$ ), we can correct a random set of $(1/2-o(1))n$ errors with high probability. For high rate codes (of degree $m-r$ for $r=o(\sqrt {m/\log m})$ ), we can correct roughly $m^{r/2}$ errors. More generally, for any integer $r$ , our algorithm can correct any error pattern in $RM(m-(2r+2),m)$ , for which the same erasure pattern can be corrected in $RM(m-(r+1),m)$ . The results above are obtained by applying recent results of Abbe, Shpilka, and Wigderson (STOC, 2015) and Kudekar et al. (STOC, 2016) regarding the ability of RM codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding RM codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the error-locating pair method of Pellikaan, Duursma, and Kotter that generalizes the Berlekamp–Welch algorithm for decoding Reed–Solomon codes.

Partial Inverses mod $m(x)$ and Reverse Berlekamp–Massey Decoding

This semi-tutorial paper introduces the partial-inverse problem for polynomials and develops its application to decoding Reed–Solomon codes and some related codes. The most natural algorithm to solve the partial-inverse problem is very similar to, but more general than, the Berlekamp–Massey algorithm. Two additional algorithms are obtained as easy variations of the basic algorithm: the first variation is entirely new, while the second variation may be viewed as a version of the Euclidean algorithm. Decoding Reed–Solomon codes (and some related codes) can be reduced to the partial-inverse problem, both via the standard key equation and, more naturally, via an alternative key equation with a new converse. Shortened and singly-extended Reed–Solomon codes are automatically included. Using the properties of the partial-inverse problem, two further key equations with attractive properties are obtained. The paper also points out a variety of options for interpolation.

On deep holes of generalized Reed-Solomon codes

Determining deep holes is an important topic in decoding Reed-Solomon codes. In a previous paper [8], we showed that the received word $u$ is a deep hole of the standard Reed-Solomon codes $[q-1, k]_q$ if its Lagrange interpolation polynomial is the sum of monomial of degree $q-2$ and a polynomial of degree at most $k-1$. In this paper, we extend this result by giving a new class of deep holes of the generalized Reed-Solomon codes.

A fast algorithm for polynomial evaluation in Reed-Solomon codec

An efficient lookup table algorithm for computing the values of message polynomials during high throughput encoding of Reed–Solomon (RS) codes is presented in this paper. The algorithm can be applied to RS codes encoders, which are based on Vandermonde matrix and the polynomial computations. The lookup table derived from the algorithm can then be applied not only to an encoder of RS codes but also to syndromes evaluation in the decoding of RS codes. By comparison with Horner’s rule, one of the advantages of utilizing this algorithm is that the table lookup operations for computing the values of the message polynomial are reduced by a factor of three. It would reduce the encoding time by fifty-percent using the linear feedback shift register to encode the (204, 188, t = 8) RS code. The algorithm can also be used to evaluate the syndromes needed in part of the RS decoder, and thus the speed is much faster than Horner’s rule. Ultimately, the proposed encoding and syndrome evaluation algorithm for RS codes can be made regular, simple and suitable for software implementations.

Syndrome Generation and Error Location Search for the Decoding of Algebraic-Geometry Codes on Plane Garcia–Stichtenoth Curves

In this paper, the basic function field of an asymptotically optimal tower of function fields by Garcia and Stichtenoth is studied, where bases of one-point linear systems can be explicitly constructed. A representation of finite GF (q 2 )-rational points on the associated plane curve is derived in this paper. This representation is exploited to extend the use of Horner's loops and the mechanism of Chien search in the decoding of Reed-Solomon codes for syndrome generation and error-location search, respectively, to algebraic-geometry codes built on the plane Garcia-Stichtenoth curve.

NOVEL PIPELINED INTERPOLATOR FOR REED–SOLOMON DECODER BASED ON LOW-COMPLEXITY CHASE DECODING

Reed–Solomon (RS) codes are widely used in digital communication and storage systems. Algebraic soft-decision (ASD) decoding of RS codes can obtain significant coding gain over hard-decision decoding. Compared with other ASD algorithms, the low-complexity chase (LCC) decoding algorithm needs less computation complexity with similar or better coding gain. To reduce the latency of the interpolation, multiple interpolators can be applied. However, the interpolator has to finish the forward interpolation (FI) which costs more iterations before it turns to the unified-backward–forward interpolator (UBFI). In this paper, FI and UBFI are carried out by two different interpolators. A novel pipelined interpolator (NPI) architecture is proposed which includes a parallel forward interpolator (PFI) and a reduced-complexity multi-interpolator (RCMI). For the (255, 239) RS code with η = 5, the interpolation latency and the efficiency will be 44% and 1.76 times of the previous design, respectively.

Error correcting codes for robust color wavelet watermarking

Abstract This article details the conception, design, development and analysis of invisible, blind and robust color image watermarking algorithms based on the wavelet transform. Using error correcting codes, the watermarking algorithms are designed to be robust against intentional or unintentional attacks such as JPEG compression, additive white Gaussian noise, low pass filter and color attacks (hue, saturation and brightness modifications). Considering the watermarking channel characterized by these attacks, repetition, Hamming, Bose Chaudhuri Hocquenghem and Reed-Solomon codes are used in order to improve the robustness using different modes and appropriate decoding algorithms. The article compares the efficiency of different type of codes against different type of attacks. To the best of our knowledge this is the first time that the effect of error-correcting codes against different attacks are detailed in a watermarking context in such a precise way: describing and comparing the effect of different classes of codes against different type of attacks. This article clearly shows that list decoding of Reed-Solomon codes using the algorithm of Sudan exhibits good performance against hue and saturation attacks. The use of error correcting codes in a concatenation mode allows the non-binary block codes to show good performance against JPEG compression, noise and brightness attacks.

On deep holes of standard Reed-Solomon codes

Determining deep holes is an important open problem in decoding Reed-Solomon codes. It is well known that the received word is trivially a deep hole if the degree of its Lagrange interpolation polynomial equals the dimension of the Reed-Solomon code. For the standard Reed-Solomon codes $[p-1, k]_p$ with $p$ a prime, Cheng and Murray conjectured in 2007 that there is no other deep holes except the trivial ones. In this paper, we show that this conjecture is not true. In fact, we find a new class of deep holes for standard Reed-Solomon codes $[q-1, k]_q$ with $q$ a prime power of $p$. Let $q \geq 4$ and $2 \leq k\leq q-2$. We show that the received word $u$ is a deep hole if its Lagrange interpolation polynomial is the sum of monomial of degree $q-2$ and a polynomial of degree at most $k-1$. So there are at least $2(q-1)q^k$ deep holes if $k \leq q-3$.

High-Throughput Interpolator Architecture for Low-Complexity Chase Decoding of RS Codes

In this paper, a high-throughput interpolator architecture for soft-decision decoding of Reed-Solomon (RS) codes based on low-complexity chase (LCC) decoding is presented. We have formulated a modified form of the Nielson's interpolation algorithm, using some typical features of LCC decoding. The proposed algorithm works with a different scheduling, takes care of the limited growth of the polynomials, and shares the common interpolation points, for reducing the latency of interpolation. Based on the proposed modified Nielson's algorithm we have derived a low-latency architecture to reduce the overall latency of the whole LCC decoder. An efficiency of at least 39%, in terms of area-delay product, has been achieved by an LCC decoder, by using the proposed interpolator architecture, over the best of the previously reported architectures for an RS(255,239) code with eight test vectors. We have implemented the proposed interpolator in a Virtex-II FPGA device, which provides 914 Mb/s of throughput using 806 slices.

Polynomial evaluation over finite fields: new algorithms and complexity bounds

An efficient evaluation method is described for polynomials in finite fields. Its complexity is shown to be lower than that of standard techniques, when the degree of the polynomial is large enough compared to the field characteristic. Specifically, if n is the degree of the polynomiaI, the asymptotic complexity is shown to be $${O(\sqrt{n})}$$ , versus O(n) of classical algorithms. Applications to the syndrome computation in the decoding of Reed-Solomon codes are highlighted.

An Interpolation Procedure for List Decoding Reed–Solomon Codes Based on Generalized Key Equations

The key step of syndrome-based decoding of Reed-Solomon codes up to half the minimum distance is to solve the so-called Key Equation. List decoding algorithms, capable of decoding beyond half the minimum distance, are based on interpolation and factorization of multivariate polynomials. This article provides a link between syndrome-based decoding approaches based on Key Equations and the interpolation-based list decoding algorithms of Guruswami and Sudan for Reed-Solomon codes. The original interpolation conditions of Guruswami and Sudan for Reed-Solomon codes are reformulated in terms of a set of Key Equations. These equations provide a structured homogeneous linear system of equations of Block-Hankel form, that can be solved by an adaption of the Fundamental Iterative Algorithm. For an $(n,k)$ Reed-Solomon code, a multiplicity $s$ and a list size $\listl$, our algorithm has time complexity \ON{\listl s^4n^2}.

A modified Guruswami–Sudan algorithm for decoding Reed–Solomon codes

We tighten a key estimate, which dictates the computational complexity of Guruswami–Sudan algorithm, on the lower bound of the degrees of freedom, and then propose a modified decoding algorithm for Reed–Solomon codes beyond half the minimum distance. The computational complexity of the modified algorithm is lower than the Guruswami–Sudan algorithm for the medium to high rate Reed–Solomon codes, and has the same asymptotic complexity as the algorithm proposed by Wu. Besides we also claim that our modified algorithm outperforms Wu's algorithm to some extent.

Soft-decision list decoding of hermitian codes

This paper proposes the first complete soft-decision list decoding algorithm for Hermitian codes based on the Koetter-Vardy's Reed-Solomon code decoding algorithm. For Hermitian codes, interpolation processes trivariate polynomials which are defined over the pole basis of a Hermitian curve. In this paper, the interpolated zero condition of a trivariate polynomial with respect to a multiplicity matrix M is redefined followed by a proof of the validity of the soft-decision scheme. This paper also introduces a new stopping criterion for the algorithm that tranforms the reliability matrix Pi to the multiplicity matrix M. Geometric characterisation of the trivariate monomial decoding region is investigated, resulting in an asymptotic optimal performance bound for the soft-decision decoder. By defining the weighted degree upper bound of the interpolated polynomial, two complexity reducing modifications are introduced for the soft-decision scheme: elimination of unnecessary interpolated polynomials and pre-calculation of the coefficients that relate the pole basis monomials to the zero basis functions of a Hermitian curve. Our simulation results and analyses show that soft-decision list decoding of Hermitian code can outperform Koetter-Vardy decoding of Reed-Solomon code which is defined in a larger finite field, but with less decoding complexity.

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

Gao's algorithm similar to Shiozaki's algorithm for decoding nonsystematic Reed-Solomon (RS) codes which is a subclass of the redundant residue polynomial codes is considered here. This algorithm computes the message polynomial directly without explicitly finding the error-locator polynomial and errorevaluator polynomial. In this letter, a simplified decoding algorithm to correct both errors and erasures is used in conjunction with Gao's algorithm for efficiently decoding RS codes. In addition, we show that the extended Gao algorithm similar to the Shiozaki-Truong-Cheung-Reed algorithm significantly reduces the decoding complexity.

Two algorithms for soft-decision decoding of reed-solomon codes, with application to multilevel coded modulations

In this paper two symbol-level soft-decision decoding algorithms for Reed-Solomon codes, derived form the ordered statistics (OS) and from the generalized minimum-distance (GMD) decoding methods, are presented and analyzed. Both the OS and the GMD algorithms are based on the idea of producing a list of candidate code words, among which the one having the larger likelihood is selected as output. We propose variants of the mentioned algorithms that allow to finely tune the size of the list in order to obtain the desired decoding complexity. The method proposed by Agrawal and Vardy for computing the error probability of the GMD algorithm is extended to our decoding methods. Examples are presented where these algorithms are applied to singly-extended Reed-Solomon codes over GF(16) used as outer codes in a 128-dimensional coded modulation scheme that attains good performance, with manageable decoding complexity.

Algebraic Soft-Decision Decoding of Reed&amp;#x2013;Solomon Codes Using Bit-Level Soft Information

The performance of algebraic soft-decision decoding of Reed-Solomon codes using bit-level soft information is investigated. Optimal multiplicity assignment strategies for algebraic soft-decision decoding (SDD) with infinite cost are first studied over erasure channels and the binary-symmetric channel. The corresponding decoding radii are calculated in closed forms and tight bounds on the error probability are derived. The multiplicity assignment strategy and the corresponding performance analysis are then generalized to characterize the decoding region of algebraic SDD over a mixed error and bit-level erasure channel. The bit-level decoding region of the proposed multiplicity assignment strategy is shown to be significantly larger than that of conventional Berlekamp-Massey decoding. As an application, a bit-level generalized minimum distance decoding algorithm is proposed. The proposed decoding compares favorably with many other Reed-Solomon SDD algorithms over various channels. Moreover, owing to the simplicity of the proposed bit-level generalized minimum distance decoding, its performance can be tightly bounded using order statistics.

Improvements on the Johnson bound for Reed–Solomon codes

For Reed–Solomon codes with block length n and dimension k , the Johnson theorem states that for a Hamming ball of radius smaller than n − n k , there can be at most O ( n 2 ) codewords. It was not known whether for larger radius, the number of codewords is polynomial. The best known list decoding algorithm for Reed–Solomon codes due to Guruswami and Sudan [Venkatesan Guruswami, Madhu Sudan, Improved decoding of Reed–Solomon and algebraic-geometry codes, IEEE Transactions on Information Theory 45 (6) (1999) 1757–1767] is also known to work in polynomial time only within this radius. In this paper we prove that when k < α n for any constant 0 < α < 1 , we can overcome the barrier of the Johnson bound for list decoding of Reed–Solomon codes (even if the field size is exponential). More specifically in such a case, we prove that for Hamming ball of radius n − n k + c (for any c > 0 ) there can be at most O ( n c 2 α ( 1 − α ) 2 + c + 2 ) number of codewords. For any constant c , we describe a polynomial time algorithm for enumerating all of them, thereby also improving on the Guruswami–Sudan algorithm. Although the improvement is modest, this provides evidence for the first time that the n − n k bound is not sacrosanct for such a high rate. We apply our method to obtain sharper bounds on a list recovery problem introduced by Guruswami and Rudra [Venkatesan Guruswami, Atri Rudra, Limits to list decoding Reed–Solomon codes, IEEE Transactions on Information Theory 52 (8) (2006) 3642–3649] where they establish super-polynomial lower bounds on the output size when the list size exceeds ⌈ n k ⌉ . We show that even for larger list sizes the problem can be solved in polynomial time for certain values of k .