Abstract
The aim of the paper is to revive and develop results of an unpublished manuscript of A.G. D'yachkov. We consider a discrete memoryless channel (DMC) and prove a theorem on the exponential expurgation bound for list decoding with fixed list size \(L\). This result is an extension of the classical exponential error probability bound for optimal codes over a DMC to the list decoding model over a DMC. As applications of this result, we consider a memoryless binary symmetric channel (BSC) and a memoryless binary asymmetric channel (Z-channel). For the both channels, we derive a lower bound on the fraction of correctable errors for zero-rate transmission over the corresponding channels under list decoding with a fixed list size \(L\) at the channel output. For the Z-channel, we obtain this bound for an arbitrary input alphabet distribution \((1-w,w)\); we also find the optimum value of the obtained bound and prove that the fraction of errors correctable by an optimal code tends to 1 as the list size \(L\) tends to infinity.
Similar content being viewed by others
References
Elias, P., List Decoding for Noisy Channels, Tech. Rep. of the Research Lab. of Electronics, MIT., Cambridge, MA, 1957, no. 335 (Reprinted from: IRE WESCON Convention Record. Part 2, pp. 99–104). Available at https://dspace.mit.edu/handle/1721.1/4484
Wozencraft, J.M., List Decoding, Quarterly Progress Report, Research Lab. of Electronics, MIT, Cambridge, MA, 1958, vol. 48, pp. 90–95.
Elias, P., Error-Correcting Codes for List Decoding, IEEE Trans. Inform. Theory, 1991, vol. 37, no. 1, pp. 5–12. https://doi.org/10.1109/18.61123
Blinovskii, V.M., Lower Bound on Error Probability in Fixed-Volume List Decoding, Probl. Peredachi Inf., 1991, vol. 27, no. 4, pp. 17–33 [Probl. Inf. Transm. (Engl. Transl.), 1991, vol. 27, no. 4, pp. 288–302]. http://mi.mathnet.ru/eng/ppi578
Blinovsky, V.M., Bounds for Codes in the Case of Finite-Volume List Decoding, Probl. Peredachi Inf., 1986, vol. 22, no. 1, pp. 11–25 [Probl. Inf. Transm. (Engl. Transl.), 1986, vol. 22, no. 1, pp. 7–19]. http://mi.mathnet.ru/eng/ppi839
D’yachkov, A.G., Expurgation Bound for Fixed Size List Decoding in a Discrete Memoryless Channel, unpublished manuscript, 1982.
Gallager, R.G., Information Theory and Reliable Communication, New York: Wiley, 1968.
Polyanskii, N. and Zhang, Y., Codes for the Z-Channel, https://arXiv.org/abs/2105.01427 [cs.IT], 2021.
Lebedev, A., Lebedev, V., and Polyanskii, N., Two-Stage Coding over the Z-Channel, https://arXiv.org/abs/2010.16362v2 [cs.IT], 2020.
Author information
Authors and Affiliations
Additional information
Translated from Problemy Peredachi Informatsii, 2021, Vol. 57, No. 4, pp. 3–23 https://doi.org/10.31857/S055529232104001X.
Rights and permissions
About this article
Cite this article
D’yachkov, A., Goshkoder, D. New Lower Bounds on the Fraction of Correctable Errors under List Decoding in Combinatorial Binary Communication Channels. Probl Inf Transm 57, 301–320 (2021). https://doi.org/10.1134/S0032946021040013
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0032946021040013