Abstract
In this paper, we consider encoding strategies for the Z-channel with noiseless feedback. We analyze the asymptotic case where the maximal number of errors is proportional to the blocklength, which goes to infinity. Without feedback, the asymptotic rate of error-correcting codes for the error fraction \(\tau \ge 1/4\) is known to be zero. It was also proved that using the feedback a non-zero asymptotic rate can be achieved for the error fraction \(\tau <1/2\). In this paper, we give an encoding strategy that achieves the asymptotic rate \((1+\tau )(1 - h(\tau /(1+\tau )))\), which is positive for all \(\tau <1\). Additionally, we state an upper bound on the maximal asymptotic rate of error-correcting codes for the Z-channel.
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Acknowledgment
Christian Deppe was supported by the Bundesministerium für Bildung und Forschung (BMBF) through Grant 16KIS1005. Vladimir Lebedev’s work was supported by the Russian Foundation for Basic Research (RFBR) under Grant No. 19-01-00364 and by RFBR and JSPS under Grant No. 20-51-50007. Georg Maringer’s work was supported by the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG) under Grant No. WA3907/4-1. Nikita Polyanskii’s research was supported in part by a German Israeli Project Cooperation (DIP) grant under grant no. KR3517/9-1.
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Deppe, C., Lebedev, V., Maringer, G., Polyanskii, N. (2020). Coding with Noiseless Feedback over the Z-Channel. In: Kim, D., Uma, R., Cai, Z., Lee, D. (eds) Computing and Combinatorics. COCOON 2020. Lecture Notes in Computer Science(), vol 12273. Springer, Cham. https://doi.org/10.1007/978-3-030-58150-3_8
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