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Concatenated Codes

Ron M. Roth

Published Feb 23, 2006• 0 citations

Abstract

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.

DOI: 10.1017/cbo9780511808968.013

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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.

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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.

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.

We consider a concatenated code with designed distance dodi

/2

, based on an outer code with distance do and an inner code with distance di. To decode the inner code, we use a Bounded Minimum Distance decoder correcting up to (di

-1

)

/2

errors. For decoding the outer code, we use a

\lambda

-Bounded Distance decoder correcting

\varepsilon

errors and

\tau

erasures if

\lambda\varepsilon+\tau \leq

do

/2

, where a real number

1dido

/(\lambda +1)$ if the number

\tau

of erasures is fixed, and (dido

/2

)∗

(1-(\frac{\lambda-1}{\lambda})^2)

for adaptive selection of

\tau

. The error-correcting radius quickly approaches dido

/2

with decreasing

\lambda

. These results can be applied e.g. when punctured Reed-Solomon outer codes are used.

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/.