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Advances in Mathematics of Communications

A weight-based characterization of the set of correctable error patterns under list-of-2 decoding

Jonas Eriksson

Published Jan 1, 2007• 0 citations
Publisher

Abstract

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 $(\tau, 2$)-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.

DOI
10.3934/amc.2007.1.331

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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(\tau, 2(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.
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/.
This paper is concerned with list decoding of 2-interleaved binary alternant codes. The principle of the proposed algorithm is based on a combination of a list decoding algorithm for (interleaved) Reed-Solomon codes and an algorithm for (non-interleaved) alternant codes. A new upper bound on the decoding radius is derived and the list size is shown to scale polynomially in the code parameters. While it remains an open problem whether this upper bound is achievable, the provided simulation results show that a decoding radius exceeding the binary Johnson radius can be achieved with a high probability of decoding success by the proposed algorithm.
We present an approach to showing that a linear code is resilient to random errors. We use this approach to obtain decoding results for both transitive codes and Reed-Muller codes. We give three kinds of results about linear codes in general, and transitive linear codes in particular. 1) We give a tight bound on the weight distribution of every transitive linear code C⊆F2NC \subseteq \mathbb{F}_2^NC⊆F2N​: Pr⁡c∈C[∣c∣=αN]≤2−(1−h(α))dim(C)\Pr_{c \in C}[|c| = αN] \leq 2^{-(1-h(α)) \mathsf{dim}(C)}Prc∈C​[∣c∣=αN]≤2−(1−h(α))dim(C). 2) We give a criterion that certifies that a linear code CCC can be decoded on the binary symmetric channel. Let Ks(x)K_s(x)Ks​(x) denote the Krawtchouk polynomial of degree sss, and let C⊥C^\perpC⊥ denote the dual code of CCC. We show that bounds on Ec∈C⊥[KεN(∣c∣)2]\mathbb{E}_{c \in C^{\perp}}[ K_{εN}(|c|)^2]Ec∈C⊥​[KεN​(∣c∣)2] imply that CCC recovers from errors on the binary symmetric channel with parameter εεε. Weaker bounds can be used to obtain list-decoding results using similar methods. One consequence of our criterion is that whenever the weight distribution of C⊥C^\perpC⊥ is sufficiently close to the binomial distribution in some interval around N2\frac{N}{2}2N​, CCC is resilient to εεε-errors. 3) We combine known estimates for the Krawtchouk polynomials with our weight bound for transitive codes, and with known weight bounds for Reed-Muller codes, to obtain list-decoding results for both these families of codes. In some regimes, our bounds for Reed-Muller codes achieve the information-theoretic optimal trade-off between rate and list size.
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.