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IEEE International Symposium on Information Theory (ISIT)

List decoding of concatenated codes: improved performance estimates

Alexander Barg, Andrew McGregor

Published Dec 23, 2004• 0 citations
Publisher

Abstract

An improved bound is proved on the list-decoding radius of a concatenated code relying upon a combination of (soft-decision) algebraic list decoding and generalized minimum distance (GMD) decoding in the outer level. This bound is further improved if the inner code is a random linear code.
DOI
10.1109/isit.2004.1365458

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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.
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.
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.
Let a binary Reed-Muller code RM(s;m) of length n be used on a memoryless channel with an input alphabet ±1 and a real-valued output ℝ. Given a received vector y in ℝn; we define its generalized distance T to any codeword c as the sum Σ|yj|taken over all positions j, in which vectors y, c have opposite signs. We then consider the list ℒTof codewords located within distance T from the received vector y and estimate the size LTof this list using the generalized Johnson bound. For any RM code RM(s,m) of fixed order s, the algorithm is proposed that performs list decoding beyond the error-correcting radius with linear complexity in length n and retrieves the code list ℒTwith complexity of order nsLTfor any decoding radius T within the generalized Johnson bound.
Recently, G. Zemor (see IEEE Trans. Inf. Theory, vol.47, p.835-7, 2001) proposed an improvement on the Sipser-Spielman analysis of expander codes (Sipser, M. and Spielman, D.A., IEEE Trans. Inf. Theory, vol.42 , p.1710-22, 1996) and presented a linear-time iterative decoder that can correct a number of errors up to approximately 1/4 the known lower bound on the minimum distance of the code. We propose an improvement on Zemor's decoder for F=GF(2), with the number of correctable errors becoming close to half the lower bound on the minimum distance. The improvement is obtained by inserting into the decoding algorithm features akin to generalized minimum distance decoding of concatenated codes.
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.