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Problems of Information Transmission

On the theory of list decoding of convolutional codes

Rolf Johannesson, Kamil Sh. Zigangirov

Published Jan 1, 1996• 3 citations
OpenAlex

Abstract

Lower bounds on the list minimum weight and list weight are obtained for ensembles of systematic and nonsystematic encoders. >From these bounds it follows that the necessary list size grows exponentially with the number of corrected errors. We obtain an expurgated upper bound on the probability of the correct path loss for the ensemble of time-invariant binary convolutional codes. The notion of an l-list weight enumerator is introduced and used to obtain an upper bound on the probability of the correct path loss for list decoding of a fixed convolutional code.

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Lower bounds on the list minimum weight and list weight are obtained for ensembles of systematic and nonsystematic encoders. >From these bounds it follows that the necessary list size grows exponentially with the number of corrected errors. We obtain an expurgated upper bound on the probability of the correct path loss for the ensemble of time-invariant binary convolutional codes. The notion of an l-list weight enumerator is introduced and used to obtain an upper bound on the probability of the correct path loss for list decoding of a fixed convolutional code.
A list decoder generates a list of more than one codeword candidates, and decoding is erroneous if the transmitted codeword is not included in the list. This decoding strategy can be implemented in a system that employs an inner error correcting code and an outer error detecting code that is used to choose the correct codeword from the list. Probability of codeword error analysis for a linear block code with list decoding is typically based on the “worst case” lower bound on the effective weights of codewords for list decoding evaluated from the weight enumerating function of the code.
For a given linear code (ensemble), upper bounds on the error probability under maximum likelihood decoding are investigated at low rates. A class of symmetric memoryless channels suitable for Bhattacharyya-type bounds, which are particularly of importance at low rates, is introduced. Over a symmetric channel, a lower bound on the error exponent is derived for a given linear code. A sufficient condition for achieving the expurgated exponent, which is the best among known error exponents at low rates, within a fixed discrepancy is given. Over a general discrete memoryless channel, the same analysis provides a lower bound on the average error exponent for an ensemble of coset codes generated by a given linear code. The bounding technique is extended to the case of generalized maximum likelihood decoding with erasure and list- decoding options.
In the list-of-L decoding of a block code the receiver of a noisy sequence lists L possible transmitted messages, and is in error only if the correct message is not on the list. Consideration is given to (n,e,L) codes, which correct all sets of e or fewer errors in a block of n bits under list-of-L decoding. New geometric relations between the number of errors corrected under list-of-1 decoding and the (larger) number corrected under list-of-L decoding of the same code lead to new lower bounds on the maximum rate of (n,e,L) codes. They show that a jammer who can change a fixed fraction p>
A lower bound on the minimum distance of convolutional polar codes is provided. The bound is obtained from the minimum weight of the generalized cosets of the codes generated by the bottom rows of the polarizing matrix. Moreover, a construction of convolutional polar subcodes is proposed, which provides improved performance under successive cancellation list decoding. For sufficiently large list size, the decoding complexity of convolutional polar subcodes appears to be lower compared with Arikan polar subcodes with the same performance. The error probability of successive cancellation list decoding of convolutional polar subcodes is lower than that of Arikan polar subcodes with the same list size.
In this paper, we study the list decoding error probability of a linear code over the erasure channel. The notion of L-incorrigible sets of a linear code is introduced to characterize its performance under list decoding. The L-incorrigible set distribution of a linear code can also be used to completely determine its decoding error probability under maximum likelihood decoding over the erasure channel. Furthermore, we show that the L-incorrigible set distribution of a linear code can be determined by its support weight distribution. Finally, the error exponent of the unsuccessful decoding probability under optimal decoding for the ensemble of all [n, nR] linear codes is determined.