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IEEE International Symposium on Information Theory (ISIT)

An Algorithm For Soft Decoding Of Block Codes

Jan Nilsson

Published Aug 24, 2005• 1 citation
Publisher

Abstract

A well known soft decoding algorithm that can make use of algebraic decoding techniques is the GMD algorithm introduced by Forney. This algorithm is based on several error-erasure decoding trails where unreliable positions are successively erased. Chase proposed a closely related family of algorithms for binary codes where unreliable positions are switched', i.e. different estimates for the positions are used in intermediate steps during the decoding process. In our algorithm, suitable for non binary codes, we try to combine erasures and switchings in a favourable way. A version of the algorithm comparable in complexity with the GMD algorithm is investigated. Simulations show that our algorithm in many cases has a somewhat better error correcting capability then the GMD algorithm.

DOI
10.1109/isit.1991.695247

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A well known soft decoding algorithm that can make use of algebraic decoding techniques is the GMD algorithm introduced by Forney. This algorithm is based on several error-erasure decoding trails where unreliable positions are successively erased. Chase proposed a closely related family of algorithms for binary codes where unreliable positions are switched', i.e. different estimates for the positions are used in intermediate steps during the decoding process. In our algorithm, suitable for non binary codes, we try to combine erasures and switchings in a favourable way. A version of the algorithm comparable in complexity with the GMD algorithm is investigated. Simulations show that our algorithm in many cases has a somewhat better error correcting capability then the GMD algorithm.
The error-correction capability and the performance of generalized minimum distance decod- ing based on threshold erasing is studied as a function of the number of decoding trials and of the allowable reliability values. Generalized Minimum Distance (GMD) decoding, as intro- duced by Forney (l), is a multitrial decoding algorithm that combines reliability information, represented by numbers in the closed interval (0,1), with algebraic decoding algorithms for error correction. In each trial, a fixed number of the most unreliable symbols in the received sequence is erased. If a cer- tain distance criterion is satisfied, Forney has shown that the transmitted codeword can be retrieved in at most (d/2) trials, where d is the Hamming distance of the code. In this work, we study a variation on Forney's scheme where, in each trial, symbols whose reliabilities are below a certain threshold are erased, and hence there is no need to sort the received symbols according to reliability values be- fore erasing as in Forney's GMD decoding. Furthermore, we consider limiting the allowable reliability values to the set R, = u~=mo/2'{1 - 2i/m}, where m = 1,2,3,. . .. This covers the simple cases of taking only hard decisions (m = l), allow- ing erasures (m = 2), or accommodating the full range (0,1) of reliability values as in Forney's GMD decoding (m + 00). Limiting the reliability values may be of importance since, in any digital implementation, the reliability values need to be quantized. Moreover, in some applications, e.g., GMD decod- ing of concatenated codes over discrete channels, the reliabil- ity values are discrete in nature (2). The maximum number of trials is kept as a parameter. Limiting the reliability values and the number of trials leads to the concept of reduced GMD decoding.
For binary linear block codes, we introduce a multiple generalized minimum distance (GMD) decoding algorithm, where GMD-like decoding is iterated around a few appropriately selected search centers. Compared with the original GMD decoding by Forney (1966), this decoding algorithm provides better error performance by increasing the number of iterations of erasure and error correction moderately. To reduce the number of iterations, we derive new effective sufficient conditions on the optimality of decoded codewords.
We present a general version of the generalized minimum distance (GMD) decoding algorithm (see Forney, G.D., Jr, 1966) that accommodates different erasing strategies. This version is called reduced GMD decoding since, depending on the erasing strategy used, it can offer a reduction in complexity over Forney's GMD decoding. We apply reduced GMD decoding to concatenated codes. In particular, we study the error correction capability and the unsuccessful decoding probability of concatenated codes with reduced GMD decoding.