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IEEE Transactions on Information Theory

On the computation of the probability of post-decoding error events for block codes (Corresp.)

Zelma McC. Huntoon, Arnold M. Michelson

Published May 1, 1977• 45 citations
Publisher

Abstract

The computation of the probability of certain post-decoding error events is formulated for block codes whose weight distributions are known. A bounded-distance decoder for group codes is assumed. Numerical examples included show post-decoding character error rates for the Golay code and for some Reed-Solomon codes and the distribution of post-decoding errors for the same Reed-Solomon codes. An extension to errors and erasures decoding is also given.

DOI
10.1109/tit.1977.1055703

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The computation of the probability of certain post-decoding error events is formulated for block codes whose weight distributions are known. A bounded-distance decoder for group codes is assumed. Numerical examples included show post-decoding character error rates for the Golay code and for some Reed-Solomon codes and the distribution of post-decoding errors for the same Reed-Solomon codes. An extension to errors and erasures decoding is also given.
A general formula is derived to obtain the decoder error probability for linear block codes. A bounded-distance decoder is assumed. It is shown that the decoder error probability for binary linear codes becomes a constant value if the weight distribution of the code is binomial-like. An exact decoder error probability for Reed-Solomon codes is obtained from this formula. Bounds on the decoder error probability for binary BCH codes are derived because the weight distributions of binary BCH codes are unknown except for some special codes. It is also shown that the decoder error probability approaches to (formula) rapidly in t-errors correcting (n, k) binary primitive BCH codes as code length n increases and the weight of word h approaches to n2.
Determining the exact decoding error probability of linear block codes is an interesting problem. For binary BCH codes, McEliece derived methods to estimate the error probability of a simple bounded distance decoding (BDD) for BCH codes. However, BDD falls short in many applications. In this work, we consider error-and-erasure decoding and its improved variants. We derive closed-form expressions for their error probabilities and validate them through simulations. Then, we illustrate their use in assessing concatenated coding schemes.
A method for estimating the block error probability of Reed-Muller codes on a binary erasure channel and a binary symmetric channel under a permutation decoding algorithm is proposed. This method is based on an analysis of the block error probability under a recursive decoding algorithm for a fixed number of input erasures or errors. Simulation results demonstrate that the proposed method allows to effectively predict the block error rate performance of Reed-Muller codes under permutation decoding with a different number of permutations used.
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/.
A coset representation of the bit error probabilities for binary linear block codes and arbitrary symmetric memoryless channels is derived. Such a representation leads to simple upper bounds on the bit error probabilities for code components expressed via 2n code spectra, where n is the code length, and to a "bounded distance decoding" algorithm. The performance of the algorithm is illustrated for the (23,12) Golay code and a particular channel having non-binary output alphabet where it attains a sub-optimum performance.