Soft-decision Decoding Of Reed-Solomon Codes Research Articles

A COMPLEXITY-REDUCED INTERPOLATION ALGORITHM FOR SOFT-DECISION DECODING OF REED-SOLOMON CODES

Soon after Lee and O'Sullivan proposed a new interpolation algorithm for algebraic soft-decision decoding of Reed-Solomon codes, there have been some attempts to apply a coordinate transformation technique to the new algorithm, with a remarkable complexity reducing effect. In this paper, a conceptually simple way of applying the transformation technique to the interpolation algorithm is proposed.

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–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.

Low-Latency Factorization Architecture for Algebraic Soft-Decision Decoding of Reed–Solomon Codes

Bivariate polynomial factorization is an important stage of algebraic soft-decision decoding of Reed-Solomon (RS) codes and contributes to a significant portion of the overall decoding latency. With the exhaustive search-based root computation method, factorization latency is dominated by the root computation step, especially for RS codes defined over very large finite fields. The root-order prediction method proposed by Zhang and Parhi only improves average latency, but does not have any effect on the worst-case latency of the factorization procedure. Thus, neither approach is well-suited for delay-sensitive applications. In this paper, a novel architecture based on direct root computation is proposed to greatly reduce the factorization latency. Direct root computation is feasible because in most practical applications of algebraic soft-decision decoding of RS codes, enough decoding gain can be achieved with a relatively low interpolation cost, which results in a bivariate polynomial with low Y-degree. Compared with existing works, not only does the new architecture have a significantly smaller worst-case decoding latency, but it is also more area efficient since the corresponding hardware for routing polynomial coefficients is eliminated.

Upper Bounds on the Number of Errors Corrected by the Koetter–Vardy Algorithm

By introducing a few simplifying assumptions we derive a simple condition for successful decoding using the Koetter-Vardy algorithm for soft-decision decoding of Reed-Solomon codes. We show that the algorithm has a significant advantage over hard decision decoding when the code rate is low, when two or more sets of received symbols have substantially different reliabilities, or when the number of alternative transmitted symbols is very small.

Algebraic Soft-Decision Decoding of Reed-Solomon Codes with Erasures on Gaussian Channels

The search for increasing the error performance of algebraic soft-decision decoding of high rate Reed-Solomon (RS) codes motivates the development of this work in an attempt to determine the ultimate error-correcting capabilities of algebraic soft-decision decoding of RS codes in Gaussian channels with erasures. It is shown through simulation that a significant performance improvement can be obtained when the unreliable bits received from channel output are declared as erased bits. An alternative method to construct the reliability matrix is developed through the mapping of the a posteriori channel probabilities in order to assign equal multiplicity for symbols with the same erased bits patterns. Conversely, it is also shown through simulation that trying to declare the unreliable symbols received from the channel as erasures does not lead to any performance gain and in some cases it would affect the decoder performance.

Architecture and Implementation of an Interpolation Processor for Soft-Decision Reed–Solomon Decoding

Reed-Solomon codes are powerful error-correcting codes that can be found in many digital communications standards. Recently, there has been an interest in soft-decision decoding of Reed-Solomon codes, incorporating reliability information from the channel into the decoding process. The Koetter-Vardy algorithm is a soft-decision decoding algorithm for Reed-Solomon codes which can provide several dB of gain over traditional hard-decision decoders. The algorithm consists of a soft-decision front end to the interpolation-based Guruswami-Sudan list decoder. The main computational task in the algorithm is a weighted interpolation of a bivariate polynomial. We propose a parallel architecture for the hardware implementation of bivariate interpolation for soft-decision decoding. The key feature is the embedding of both a binary tree and a linear array into a 2-D array processor, enabling fast polynomial evaluation operations. An field-programmable gate array interpolation processor was implemented and demonstrated at a clock frequency of 23 MHz, corresponding to decoding rates of 10-15 Mb/s

High-Speed Interpolation Architecture for Soft-Decision Decoding of Reed–Solomon Codes

Algebraic soft-decision decoding of Reed-Solomon (RS) codes delivers promising coding gains over conventional hard-decision decoding. The most computationally demanding step in soft-decision decoding of RS codes is bivariate polynomial interpolation. In this paper, we present a hybrid data format-based interpolation architecture that is well suited for high-speed implementation of the soft-decision decoders. It will be shown that this architecture is highly scalable and can be extensively pipelined. It also enables maximum overlap in time for computations at adjacent iterations. It is estimated that the proposed architecture can achieve significantly higher throughput than conventional designs with equivalent or lower hardware complexity

Exponential Error Bounds for Algebraic Soft-Decision Decoding of Reed–Solomon Codes

Algebraic soft-decision decoding of Reed-Solomon codes is a promising technique for exploiting reliability information in the decoding process. While the algorithmic aspects of the decoding algorithm are reasonably well understood and, in particular, complexity is polynomially bounded in the length of the code, the performance analysis has relied almost entirely on simulation results. Analytical exponential error bounds that can be used to tightly bound the performance of Reed-Solomon codes under algebraic soft-decision decoding are presented in this paper. The analysis is used in a number of examples and several extensions and consequences of the results are presented.

Iterative algebraic soft-decision list decoding of Reed-Solomon codes

In this paper, we present an iterative soft-decision decoding algorithm for Reed-Solomon codes offering both complexity and performance advantages over previously known decoding algorithms. Our algorithm is a list decoding algorithm which combines two powerful soft decision decoding techniques which were previously regarded in the literature as competitive, namely, the Koetter-Vardy algebraic soft-decision decoding algorithm and belief-propagation based on adaptive parity check matrices, recently proposed by Jiang and Narayanan. Building on the Jiang-Narayanan algorithm, we present a belief-propagation based algorithm with a significant reduction in computational complexity. We introduce the concept of using a belief-propagation based decoder to enhance the soft-input information prior to decoding with an algebraic soft-decision decoder. Our algorithm can also be viewed as an interpolation multiplicity assignment scheme for algebraic soft-decision decoding of Reed-Solomon codes.