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.
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.
A concatenated code $\mathcal{C} $ based on an inner code with Hamming distance di and an outer code with Hamming distance do is considered. An outer decoder that corrects ε errors and θ erasures with high probability if λε+θ≤do−1, where a real number 1<λ≤2 is the trade-off rate between errors and erasures for this decoder is used. In particular, an outer l-punctured RS code, i.e., a code over the field Fql of length no<q with locators taken from the sub-field Fq, where l∈{1,2,…} is considered. In this case, the trade-off is given by λ=1+1/l. An m-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) to the whole given range of λ. For the fixed erasing strategy the error correcting radius approaches ρF≈2dido(1−2l−m) for large do. For the adaptive erasing strategy, the error correcting radius ρA≈2dido(1−l−2m) quickly approaches dido/2 if l or m grows. The minimum number of trials required to reach an error correcting radius dido/2 is mA=21(logld+1). This means that 2 or 3 trials are sufficient in many practical cases if l>1.
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 λ-Bounded Distance decoder correcting ε errors and τ erasures if λε+τ≤do−1, where a real number 1dido/(\lambda +1)$ if the number τ of erasures is fixed, and (dido/2)∗(1−(λλ−1)2) for adaptive selection of τ. The error-correcting radius quickly approaches dido/2 with decreasing λ. These results can be applied e.g. when punctured Reed-Solomon outer codes are used.
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.
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.
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.
A new bounded-distance (BD) decoding algorithm is presented for binary linear (n, k, d) block codes on additive white Gaussian noise channels. The algorithm is based on the generalized minimum distance (GMD) decoding algorithm of Forney (1989) using the acceptance criterion of Taipale and Pursley (GMD/TP) proposed in 1991. It is shown that the GMD/TP decoding algorithm is a BD decoding algorithm with effective error coefficient (/sup n//sub d/). It is also shown that the decision regions of GMD/TP are good inner approximations of those of full GMD decoding, and therefore full GMD decoding is BD and has an effective error coefficient that is well approximated by (/sup n//sub d/). Moreover, by adding a d-erasure-correction step to GMD decoding, the effective error coefficient can be reduced to A/sub d/, the number of minimum-weight codewords, which is the same as the effective error coefficient of maximum-likelihood decoding. The decoding algorithm is mainly based on algebraic errors-and-erasures decoding and therefore has polynomial rather than exponential complexity.