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Asilomar Conference on Signals, Systems and Computers (ACSSC)

A Chernoff type error bound for algebraic soft-decision decoding of Reed Solomon codes

N. Ratnakar, R. Koetter

Published Dec 22, 2003• 3 citations
Publisher

Abstract

We present exponential upper bounds on the performance of algebraic soft-decision decoding algorithm in the decoding of Reed-Solomon codes for a symmetric channel. The symmetric channels are useful for computing the performance of the algebraic soft-decision decoding algorithm in regimes where the probability of error is low. In such regimes, these bounds are of particular interest since it is impractical to run simulations. The bounds are Chernoff-type error bounds.

DOI
10.1109/acssc.2002.1197167

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We present exponential upper bounds on the performance of algebraic soft-decision decoding algorithm in the decoding of Reed-Solomon codes for a symmetric channel. The symmetric channels are useful for computing the performance of the algebraic soft-decision decoding algorithm in regimes where the probability of error is low. In such regimes, these bounds are of particular interest since it is impractical to run simulations. The bounds are Chernoff-type error bounds.
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Error exponents are studied for recursive and majority decoding of general Reed-Muller (RM) codesRM(r,m) used on the additive white Gaussian noise (AWGN) channels. Both algorithms have low complexity and correct many error patterns whose weight exceeds half the code distance. Decoding consists of multiple consecutive steps, which repeatedly recalculate the input symbols and determine different information symbols using soft-decision majority voting. For any codeRM(r,m), we estimate the probabilities of the information symbols obtained in these recalculations and derive the analytical upper bounds for the block error rates of the recursive and majority decoding. In the case of a low noise, we also obtain the lower bounds and show that the upper bounds are tight. For a higher noise, these bounds closely approach our simulation results.
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.
An upper bound on the effective error coefficient of a two-level code with two-stage decoding is presented. This bound provides a guideline for constructing two-level codes to achieve a good trade-off between the error performance and decoding complexity. Based on this bound, good two-level decompositions of some Reed-Muller codes for two-stage decoding are found. Simulation results on the error performances of some Reed-Muller codes of lengths up to 64 with two-stage soft-decision suboptimum decoding based on their two-level decompositions are given.>