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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

A Soft-Decision Recursive Decoding Algorithm Using Iterative Bounded-Distance Decoding for u|u+v Codes

Hitoshi Tokushige

Published Jan 1, 2014• 0 citations
Publisher

Abstract

A soft-decision recursive decoding algorithm (RDA) for the class of the binary linear block codes recursively generated using a u|u+v-construction method is proposed. It is well known that Reed-Muller (RM) codes are in this class. A code in this class can be decomposed into left and right components. At a recursive level of the RDA, if the component is decomposable, the RDA is performed for the left component and then for the cosets generated from the left decoding result and the right component. The result of this level is obtained by concatenating the left and right decoding results. If the component is indecomposable, a proposed iterative bounded-distance decoding algorithm is performed. Computer simulations were made to evaluate the RDA for RM codes over an additive white Gaussian-noise channel using binary phase-shift keying modulation. The results show that the block error rates of the RDA are relatively close to those of the maximum-likelihood decoding for the third-order RM code of length 26 and better than those of the Chase II decoding for the third-order RM codes of length 26 and 27, and the fourth-order RM code of length 28.

DOI
10.1587/transfun.e97.a.996

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A soft-decision recursive decoding algorithm (RDA) for the class of the binary linear block codes recursively generated using a u|u+v-construction method is proposed. It is well known that Reed-Muller (RM) codes are in this class. A code in this class can be decomposed into left and right components. At a recursive level of the RDA, if the component is decomposable, the RDA is performed for the left component and then for the cosets generated from the left decoding result and the right component. The result of this level is obtained by concatenating the left and right decoding results. If the component is indecomposable, a proposed iterative bounded-distance decoding algorithm is performed. Computer simulations were made to evaluate the RDA for RM codes over an additive white Gaussian-noise channel using binary phase-shift keying modulation. The results show that the block error rates of the RDA are relatively close to those of the maximum-likelihood decoding for the third-order RM code of length 26 and better than those of the Chase II decoding for the third-order RM codes of length 26 and 27, and the fourth-order RM code of length 28.
Soft-decision decoding is considered for general Reed-Muller (RM) codes of length n and distance d used over a memoryless channel. A recursive decoding algorithm is designed and its decoding threshold is derived for long RM codes. The algorithm has complexity of order nlnn and corrects most error patterns of the Euclidean weight of order radicn/lnn, instead of the decoding threshold radicd/2 of the bounded distance decoding. Also, for long RM codes of fixed rate R, the new algorithm increases 4/pi times the decoding threshold of its hard-decision counterpart
Soft-decision decoding on a memoryless channel is considered for the first-order Reed-Muller codes RM (1, m) of length 2m. We assume that different positions j of the received binary vector y can be corrupted by the errors of varying weight wj. The generalized Hamming distance between vector y and any binary vector c is then defined as the sum of weighted differences wj|yj- cj| taken over all n positions. We obtain a tight upper bound LTon the number of codewords located within generalized Hamming distance T from vector y, and design a decoding algorithm that outputs this list of codewords with complexity O (n ln2LT). In particular, all possible error weights wjequal 1 if this combinatorial model is applied to a binary symmetric channel. In this case, the well known Green algorithm performs full maximum likelihood decoding of RM (1, m) and requires O (n ln2n) bit operations, whereas the Litsyn-Shekhovtsov algorithm operates within the bounded-distance decoding radius n/4-1 with linear complexity O(n). We close the performance-complexity gap between the two algorithms. Namely, for any fixed (0, ½), our algorithm outputs the complete list of codewords within the decoding radius n(½-) with linear complexity of order n ln2.
Recursive list decoding is considered for Reed-Muller (RM) codes. The algorithm repeatedly relegates itself to the shorter RM codes by recalculating the posterior probabilities of their symbols. Intermediate decodings are only performed when these recalculations reach the trivial RM codes. In turn, the updated lists of most plausible codewords are used in subsequent decodings. The algorithm is further improved by using permutation techniques on code positions and by eliminating the most error-prone information bits. Simulation results show that for all RM codes of length 256 and many subcodes of length 512, these algorithms approach maximum-likelihood (ML) performance within a margin of 0.1 dB. As a result, we present tight experimental bounds on ML performance for these codes.
Recursive list decoding is considered for Reed-Muller (RM) codes. The algorithm repeatedly relegates itself to the shorter RM codes by recalculating the posterior probabilities of their symbols. Intermediate decodings are only performed when these recalculations reach the trivial RM codes. In turn, the updated lists of most plausible codewords are used in subsequent decodings. The algorithm is further improved by using permutation techniques on code positions and by eliminating the most error-prone information bits. Simulation results show that for all RM codes of length 256 and many subcodes of length 512, these algorithms approach maximum-likelihood (ML) performance within a margin of 0.1 dB. As a result, we present tight experimental bounds on ML performance for these codes
The Plotkin construction combines two codes to a code of doubled length. It can be applied recursively. The class of Reed-Muller (RM) codes is a particular example. Exploiting a property of the code words constructed by the recursive Plotkin construction, we present novel soft-decision decoders. These are based on the decoding of hidden code words which are inherent contained in the constructed code words and can be uncovered by adding particular parts of the overall code word. The main idea is to use more than one decoding variant where each variant starts with the decoding of a different hidden code word. Given the decision of this first hidden code word allows error cancellation for the remaining decoding. The final decoding decision selects the best of the decisions of the used variants. The more variants are used the closer the performance gets to the maximum-likelihood (ML) decoding performance. This is verified by an ML-bound for the cases where the ML performance is not known. The decoding algorithms use only additions, comparisons, and sign operations. Further, due to the recursive structure, only relatively short codes have to be decoded, thus, the decoding complexity is very low. We also present a new decoder for first-order RM codes with low complexity. In addition, we introduce two novel classes of half-rate codes based on recursive Plotkin constructions with RM codes. We show that the novel soft decision decoders can also be applied to recursive Plotkin constructions with BCH codes and to particular classes of generalized concatenated codes (GCC).
New soft- and hard decision decoding algorithms are presented for general Reed-Muller codes $\left\{\genfrac{}{}{0pt}{}{m}{r}\right\} $ of length 2m2^{m}2m and distance 2m−r2^{m-r}2m−r. We use Plotkin (u,u+v)(u,u+v)(u,u+v) construction and decompose code $\left\{\genfrac{}{}{0pt}{}{m}{r}\right\} $ onto subblocks $u\in\left\{\genfrac{}{}{0pt}{}{m-1}{r}\right\} $ and v∈{m−1r−1}.v\in\left\{\genfrac{}{}{0pt}{}{m-1}{r-1}\right\} .v∈{r−1m−1​}. In decoding, we first try to find a subblock vvv from the better protected code and then proceed with the block uuu. The likelihoods of the received symbols are recalculated in a way similar to belief propagation. Thus, decoding is relegated to the two constituent codes. We repeat this recursion and execute decoding only at the end nodes $\left\{\genfrac{}{}{0pt}{}{j}{1}\right\} $ and $\left\{\genfrac{}{}{0pt}{}{j}{j-1}\right\} $. The overall complexity has low order of nlog⁡n.n\log n.nlogn. It is shown that this decoding substantially outperforms other algorithms of polynomial complexity known for RM codes. In particular, for medium and high code rates, the algorithm corrects most error patterns of weight dln⁡d/2.d\ln d/2.dlnd/2.