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/.
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/.
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.
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.
List decoding of block codes is an alternative approach to thedecoding problem with appealing qualities. The fairly recent development of efficient algorithms for list decoding of Reed-Solomon codes spur new fuel to the study of this decoding strategy. In this paper we give a weight-based characterization of the set of correctable error patterns under list-of-2 ecodingof (τ,2)-list-decodable linear codes with known weight distribution. We apply our characterization of the set of correctable error patterns to a few codes in a family of low-rate list-of-2 decodable Reed-Solomon codes. We study the increase in error-correction performance obtained in a symmetric AWGN channel by using list-of-2 decoding instead of traditional decoding for these codes. Some simulation results for list-of-2 decoding on QAM channels using the Guruswami-Sudan algorithm for decoding of Reed-Solomon codes are alsopresented.
We present a decoding algorithm for concatenated codes when the outer code is a Reed-Solomon code and the inner code is arbitrary. "Soft" information on the reliability of various symbols is passed by the inner decodings and exploited in the Reed-Solomon decoding. This is the first analysis of such a soft algorithm that works for arbitrary inner codes; prior analyses could only, handle some special inner codes. Crucial to our analysis is a combinatorial result on the coset weight distribution of codes given only its minimum distance. Our result enables us to decode essentially up to the "Johnson radius" of a concatenated code when the outer distance is large (the Johnson radius is the "a priori list decoding radius" of a code as a function of its distance). As a consequence, we are able to present simple and efficient constructions of q-ary linear codes that are list decodable up to a fraction (1 - 1/q - /spl epsiv/) of errors and have rate /spl Omega/(/spl epsiv//sup 6/). Codes that can correct such a large fraction of errors have found numerous complexity-theoretic applications. The previous constructions of linear codes with a similar rate used algebraic-geometric codes and thus suffered from a complicated construction and slow decoding.
Reed-Muller codes encode an m-variate polynomial of degree r by evaluating it on all points in {0,1}m. We denote this code by RM(m,r). The minimal distance of RM(m,r) is 2m−r and so it cannot correct more than half that number of errors in the worst case. For random errors one may hope for a better result. In this work we give an efficient algorithm (in the block length n=2m) for decoding random errors in Reed-Muller codes far beyond the minimal distance. Specifically, for low rate codes (of degree r=o(m)) we can correct a random set of (1/2−o(1))n errors with high probability. For high rate codes (of degree m−r for r=o(m/logm)), we can correct roughly mr/2 errors. More generally, for any integer r, our algorithm can correct any error pattern in RM(m,m−(2r+2)) for which the same erasure pattern can be corrected in RM(m,m−(r+1)). The results above are obtained by applying recent results of Abbe, Shpilka and Wigderson (STOC, 2015), Kumar and Pfister (2015) and Kudekar et al. (2015) regarding the ability of Reed-Muller codes to correct random erasures. The algorithm is based on solving a carefully defined set of linear equations and thus it is significantly different than other algorithms for decoding Reed-Muller codes that are based on the recursive structure of the code. It can be seen as a more explicit proof of a result of Abbe et al. that shows a reduction from correcting erasures to correcting errors, and it also bares some similarities with the famous Berlekamp-Welch algorithm for decoding Reed-Solomon codes.