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Birkhäuser Basel eBooks

On the Error Exponents of Reliability-Order-Based Decoding Algorithms for Linear Block Codes

Yuansheng Tang

Published Jan 1, 2004• 2 citations
Publisher

Abstract

In this paper, for any reliability-order-based decoding algorithm of linear block codes, we prove that the error exponent and the squared error-correction radius are equal to each other. A known method for computing the squared error-correction radius is improved. The improved method can also be used further to compute the effective error coefficient of the reliabilityorder-based decoding algorithms

DOI
10.1007/978-3-0348-7865-4_21

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In this paper, for any reliability-order-based decoding algorithm of linear block codes, we prove that the error exponent and the squared error-correction radius are equal to each other. A known method for computing the squared error-correction radius is improved. The improved method can also be used further to compute the effective error coefficient of the reliabilityorder-based decoding algorithms
This paper presents a unified method for evaluating the error-correction radii of many reliability-based soft-decision decoding algorithms for binary linear block codes. Based on this unified method, these decoding algorithms as well as their potential improvements can be compared directly. The error-correction radius for each of these decoding algorithms is determined by finding the closest point on the boundary of the decision region of the soft-decision decoder to the transmitted signal sequence. It is shown that this problem can be formulated as a constrained optimization problem. Although no general closed-form expression is possible, a simple algorithm that always converges to the optimum solution of this optimization problem is presented. Based on these results, the error-correction radii of some well-known reliability-based soft-decision decoding algorithms are revisited and compared.
In this correspondence, we consider the decoding of binary block codes over the additive white Gaussian noise (AWGN) channel with binary phase-shift keying (BPSK) signaling. By a reliability-order-based decoding algorithm (ROBDA), we mean a soft-decision decoding algorithm which decodes to the best (most likely) codeword of the form that is the sum of the hard-decision tuple and an error pattern in a set determined only by the order of the reliabilities of the hard decisions. Examples of ROBDAs include many well-known decoding algorithms, such as the generalized-minimum-distance (GMD) decoding algorithm, Chase decoding algorithms, and the reliability-based decoding algorithms proposed by Fossorier and Lin. It is known that the squared error-correction-radii of ROBDAs can be computed from the minimal squared Euclidean distances (MSEDs) between the all-one sequence and the polyhedra corresponding to the error patterns. For the computation of such MSEDs, we give a new method which is more compact than the one proposed by Fossorier and Lin. These results are further used to show that any bounded-distance ROBDA is asymptotically optimal: The ratio between the probability of decoding error of a bounded-distance ROBDA and that of the maximum-likelihood (ML) decoding approaches 1 when the signal-to-noise ratio (SNR) approaches infinity, provided that the minimum Hamming distance of the code is greater than 2.
In this correspondence, various aspects of reliability-based syndrome decoding of binary codes are investigated. First, it is shown that the least reliable basis (LRB) and the most reliable basis (MRB) are dual of each other. By exploiting this duality, an algorithm performing maximum-likelihood (ML) soft-decision syndrome decoding based on the LRB is presented. Contrarily to previous LRB-based ML syndrome decoding algorithms, this algorithm is more conveniently implementable for codes whose codimension is not small. New sufficient conditions for optimality are derived. These conditions exploit both the ordering associated with the LRB and the structure of the code considered. With respect to MRR-based sufficient conditions, they present the advantage of requiring no soft information and thus can be preprocessed and stored. Based on these conditions, low-complexity soft-decision syndrome decoding algorithms for particular classes of codes are proposed. Finally, a simple algorithm is analyzed. After the construction of the LRB, this algorithm computes the syndrome of smallest Hamming weight among o(K/sup i/) candidates, where K is the dimension of the code, for an order i of reprocessing. At practical bit-error rates, for codes of length N/spl les/128, this algorithm always outperforms any algebraic decoding algorithm capable of correcting up to t+1 errors with an order of reprocessing of at most 2, where t is the error-correcting capability of the code considered.
This correspondence presents a hybrid reliability-based decoding algorithm which combines the reprocessing method based on the most reliable basis and a generalized Chase-type algebraic decoder based on the least reliable positions. It is shown that reprocessing with a simple additional algebraic decoding effort achieves significant coding gain. For long codes, the order of reprocessing required to achieve asymptotic optimum error performance is reduced by approximately 1/3. This significantly reduces the computational complexity, especially for long codes. Also, a more efficient criterion for stopping the decoding process is derived based on the knowledge of the algebraic decoding solution.
Introduction: In modern systems for communication, data storage and processing the error-correction capability of codes are estimated for memoryless channels. In real channels the noise is correlated, which leads to grouping error in bursts. A traditional method to fight this phenomenon is channel decorrelation, which does not allow developing of coding schemes, mostly utilizing the channel capacity. Thus the development of bursts decoding algorithms for arbitrary linear codes is the actual task. Purpose: To develop a single error burst decoding algorithm for linear codes, to estimate the decoding error probability and computational complexity. Results: Two approaches are proposed to burst error correction. The first one is based on combining the window sliding modification of well-known bit-flipping algorithm with preliminary analysis of the structure of parity check matrix. The second one is based on the recursive procedure of constructing the sequence of certain events which, in the worst case, performs the exhaustive search of error bursts, but in many cases the search may be significantly decreased by using the proposed heuristics. The proposed recursive decoding algorithm allows a guaranteed correction of any single error bursts within burst-correction capability of the code, and in many cases beyond the burst-correction capability. The complexity of this algorithm is significantly lower than that of a bit flipping algorithm if the parity-check matrix of the code is sparse enough. An alternative hybrid decoding algorithm is proposed utilizing the bit-flipping approach and showing the error probability and completion time comparable to the recursive algorithm, however, in this case the possibility of a guaranteed burst correction hardly can be proved. Practical relevance: The proposed decoding methods may be used in modern and perspective communication systems, allowing energy saving and increasing reliability of data transmission by better error performance and computational complexity.