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IEEE International Symposium on Information Theory (ISIT)

Achieving the optimal guaranteed error correction rate in a simple concatenated coding scheme with single-trial decoding

Jos H. Weber, Khaled Abdel-Ghaffar

Published Nov 27, 2002• 0 citations
Publisher

Abstract

For a concatenated coding scheme using a single inner code, a single outer code, and an optimal single-trial decoding strategy, we study the guaranteed error correction rate, i.e., the number of correctable errors per transmitted symbol. In particular, it is investigated whether the optimal guaranteed error correction rate can be achieved by finite length codes or only be approached by an infinite sequence of codes of increasing lengths. It is shown that optimizing the guaranteed error correction rate for a given inner or outer code does not necessarily mean pushing the code rate to zero. Furthermore, to optimize the guaranteed error correction rate over all inner and outer codes of fixed dimensions and alphabets, the code rate of one (but not both) of these two codes should be pushed to zero.

DOI
10.1109/isit.1998.708940

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For a concatenated coding scheme using a single inner code, a single outer code, and an optimal single-trial decoding strategy, we study the guaranteed error correction rate, i.e., the number of correctable errors per transmitted symbol. In particular, it is investigated whether the optimal guaranteed error correction rate can be achieved by finite length codes or only be approached by an infinite sequence of codes of increasing lengths. It is shown that optimizing the guaranteed error correction rate for a given inner or outer code does not necessarily mean pushing the code rate to zero. Furthermore, to optimize the guaranteed error correction rate over all inner and outer codes of fixed dimensions and alphabets, the code rate of one (but not both) of these two codes should be pushed to zero.
A concatenated code $\mathcal{C} $ based on an inner code with Hamming distance did^idi and an outer code with Hamming distance dod^odo is considered. An outer decoder that corrects ε\varepsilonε errors and θ\thetaθ erasures with high probability if λε+θ≤do−1,\lambda \varepsilon + \theta \le d^o - 1,λε+θ≤do−1, where a real number 1<λ≤21<\lambda\le 21<λ≤2 is the trade-off rate between errors and erasures for this decoder is used. In particular, an outer lll-punctured RS code, i.e., a code over the field Fql\mathbb{F}_{q^{l }}Fql​ of length no<qn^{o} < qno<q with locators taken from the sub-field Fq\mathbb{F}_{q}Fq​, where l∈{1,2,…}l\in \{1,2,\ldots\}l∈{1,2,…} is considered. In this case, the trade-off is given by λ=1+1/l\lambda=1+1/lλ=1+1/l. An mmm-trial decoder, where after inner decoding, in each trial we erase an incremental number of symbols and decode using the outer decoder is proposed. The optimal erasing strategy and the error correcting radii of both fixed and adaptive erasing decoders are given. Our approach extends results of Forney and Kovalev (obtained for λ=2\lambda=2λ=2) to the whole given range of λ\lambdaλ. For the fixed erasing strategy the error correcting radius approaches ρF≈dido2(1−l−m2)\rho_F\approx\frac{d^i d^o}{2}(1-\frac{l^{-m}}{2})ρF​≈2dido​(1−2l−m​) for large dod^odo. For the adaptive erasing strategy, the error correcting radius ρA≈dido2(1−l−2m)\rho_A\approx\frac{d^i d^o}{2}(1-l^{-2m})ρA​≈2dido​(1−l−2m) quickly approaches dido/2d^i d^o/2dido/2 if lll or mmm grows. The minimum number of trials required to reach an error correcting radius dido/2d^i d^o/2dido/2 is mA=12(log⁡ld+1)m_A=\frac{1}{2}\left(\log_ld+1\right)mA​=21​(logl​d+1). This means that 222 or 333 trials are sufficient in many practical cases if l>1l>1l>1.
We consider a concatenated code with designed distance dodi/2/2/2, 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−1-1−1)/2/2/2 errors. For decoding the outer code, we use a λ\lambdaλ-Bounded Distance decoder correcting ε\varepsilonε errors and τ\tauτ erasures if λε+τ≤\lambda\varepsilon+\tau \leqλε+τ≤do−1-1−1, where a real number 1dido1dido1dido/(\lambda +1)$ if the number τ\tauτ of erasures is fixed, and (dido/2/2/2)∗(1−(λ−1λ)2)(1-(\frac{\lambda-1}{\lambda})^2)(1−(λλ−1​)2) for adaptive selection of τ\tauτ. The error-correcting radius quickly approaches dido/2/2/2 with decreasing λ\lambdaλ. These results can be applied e.g. when punctured Reed-Solomon outer codes are used.
Concatenated codes are effectively used in different approaches of error-correcting theory. We suggest new concatenated code construction that is composed of outer LDPC code and a particular designed inner LDPC code. Special optimization procedure was developed to obtain inner code with specific error-correcting characteristics. We apply the decoder of woven codes to this construction, while reducing the complexity of decoding procedure compared to a classic decoder of LDPC code. Furthermore, we improve the performance of our scheme with puncturing the code symbols. The resulting concatenated construction outperforms low-rate LDPC from the 5G standard. Besides this, we demonstrate the performance of code with a special parity-check matrix consisting of parity-check matrices of inner and outer codes from our suggested construction.
We investigate threshold-based multi-trial decoding of concatenated codes with an inner Maximum-Likelihood decoder and an outer error/erasure ℓ+1/ℓ-extended Bounded Distance decoder, i.e. a decoder which corrects ε errors and τ erasures if ℓ+1/ ℓε+τ≤ do-1, where dois the minimum distance of the outer code and ℓ ∈ ℕ\{0}. This is a generalization of Forney's GMD decoding, which was considered only for ℓ = 1, i.e. outer Bounded Minimum Distance decoding. One important example for ℓ+1/ℓ-extended Bounded Distance decoders is decoding of ℓ-Interleaved Reed-Solomon codes. Our main contribution is a threshold location formula, which allows to optimally erase unreliable inner decoding results, for a given number of decoding trials and parameter ℓ. Thereby, the term optimal means that the residual codeword error probability of the concatenated code is minimized. We give an estimation of this probability for any number of decoding trials.
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.