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IEEE International Symposium on Information Theory (ISIT)

Polynomial of correctable patterns of product codes

Nicolas Sendrier

Published Dec 17, 2002• 0 citations
Publisher

Abstract

Product codes have a poor minimum distance, but an efficient low-complexity decoding algorithm. To measure the performance of a decoder, like Reddy-Robinson's for product codes, which decode error patterns beyond half the minimum distance, we use the notion of equivalent diameter of the decoding region [4]. We produce here a bound for the performance of product code which is, for the considered example, far below the simulated performance but still above the performance of known BCH codes with same parameters.>

DOI
10.1109/isit.1994.394985

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Product codes have a poor minimum distance, but an efficient low-complexity decoding algorithm. To measure the performance of a decoder, like Reddy-Robinson's for product codes, which decode error patterns beyond half the minimum distance, we use the notion of equivalent diameter of the decoding region [4]. We produce here a bound for the performance of product code which is, for the considered example, far below the simulated performance but still above the performance of known BCH codes with same parameters.>
In this chapter, we continue the discussion on concatenated codes, which was initiated in Section 5.4. The main message to be conveyed in this chapter is that by using concatenation, one can obtain codes with favorable asymptotic performance—in a sense to be quantified more precisely—while the complexity of constructing these codes and decoding them grows polynomially with the code length. We first present a decoding algorithm for concatenated codes, due to Forney. This algorithm, referred to as a generalized minimum distance (in short, GMD) decoder, corrects any error pattern whose Hamming weight is less than half the product of the minimum distances of the inner and outer codes (we recall that this product is a lower bound on the minimum distance of the respective concatenated code). A GMD decoder consists of a nearest-codeword decoder for the inner code, and a combined error–erasure decoder for the outer code. It then enumerates over a threshold value, marking the output of the inner decoder as erasure if that decoder returns an inner codeword whose Hamming distance from the respective received sub-word equals or exceeds that threshold. We show that under our assumption on the overall Hamming weight of the error word, there is at least one threshold for which the outer decoder recovers the correct codeword. If the outer code is taken as a GRS code, then a GMD decoder has an implementation with time complexity that is at most quadratic in the length of the concatenated code.
The polynomial of correctable patterns is defined In [l] as the weight enumerator of the set of error patterns correctable by a given decoding algorithm. The polynomials of uncorrectable and miscorrected patterns can be defined as well ([5] aid (61) These polynomials allow a compact representation of a decodiilg alient to compute the coriection probabillty and ability thiougli a memoryless symetric channel. Our purpose heie is to compute the polynomials of correctable patThese results are geneialised in [5] and [SI for erasure channels. terns of concatenated codes for dlffeient decoding algorithms. I) We give the weight distribution of the erior patterns correctable by the standard decoding algorithm. * We give bounds for the weight distribution of the error patterns correct able by Block- Zyablov algorithm. This new method for evaluating concatenated codes will thus provide to evaluate the standard algorithm. It will also give a with precision the performances of the Block-Zyablov decoding algorithm which needed, up to llow, a (much more expensive) simulation. ill compute the decoding performances of the rdstrom-Robinson ([4, p. 731) inlier code, and a Reed-Solomon (255,223,33) outer code over F2~6.
The polynomial of correctable patterns is defined in [1] as the weight enumerator of the set of error patterns correctable by a given decoding algorithm. The polynomials of uncorrectable and miscorrected patterns can redefined as well ([5] and [6]). These polynomials allow a compact representation of a decoding algorithm which is sufficient to compute the correction probability and the miscorrection probability through a memoryless symetric channel. These results are generalized in [5] and [6] for erasure channels. Our purpose here is to compute the polynomials of correctable patterns of concatenated codes for different decoding algorithms. We give the weight distribution of the error patterns correctable by the standard decoding algorithm. We give bounds for the weight distribution of the error patterns correctable by Block- Zyablov algorithm. This new method for evaluating concatenated codes will thus provide an efficient way to evaluate the standard algorithm. It will also give a way to evaluate with precision the performances of the Block- Zyablov decoding algorithm which needed, up to now, a (much more expensive) simulation. As an example, we will compute the decoding performances of the concatenation of the Nordstrom-Robinson ([4, p. 73]) inner code, and a Reed-Solomon (255,223,33) outer code over F/sub 256/.
A digital communication system may transmit messages consisting of thousands or even millions of bits. While one can always break a long message into short blocks for encoding, in principle, a single, long block code will give better performance because it will protect against both error patterns in which the errors are clustered and error patterns in which the errors are scattered throughout the message. Therefore, there are many occasions where good codes of very long blocklength can be used. Although short binary cyclic codes can be quite good, the known long binary cyclic codes have a small minimum distance. Codes of large blocklength with a much larger minimum distance do in principle exist, though we know very little about these more powerful codes, nor do we know how to find them. Despite more than fifty years of intense effort, codes of large blocklength and large minimum distance, both binary and nonbinary, still elude us. Even if such codes were found, it may be that their decoding would be too complex. Accordingly, the most successful constructions for codes of large blocklength for many applications combine codes of small blocklength into more elaborate structures. We call such structures composite codes . The elementary codes from which composite codes are formed can then be called basic codes .