Two-Stage Coding Over the Z-Channel

DOI10.1109/TIT.2022.3141082zbMATH Open1497.94048arXiv2010.16362MaRDI QIDQ5088385

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Publication date: 13 July 2022

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Abstract: In this paper, we discuss two-stage encoding algorithms capable of correcting a fraction of asymmetric errors. Suppose that the encoder transmits $n$ binary symbols $(x_1,ldots,x_n)$ one-by-one over the Z-channel, in which a 1 is received only if a 1 is transmitted. At some designated moment, say $n_1$, the encoder uses noiseless feedback and adjusts further encoding strategy based on the partial output of the channel $(y_1,ldots,y_{n_1})$. The goal is to transmit error-free as much information as possible under the assumption that the total number of errors inflicted by the Z-channel is limited by $aun$, $0<au<1$. We propose an encoding strategy that uses a list-decodable code at the first stage and a high-error low-rate code at the second stage. This strategy and our converse result yield that there is a sharp transition at $au=maxlimit{s}_{0<w<1}fracw+w^{3}1+4w^{3}approx0.44$ from positive rate to zero rate for two-stage encoding strategies. As side results, we derive bounds on the size of list-decodable codes for the Z-channel and prove that for a fraction $1/4+epsilon$ of asymmetric errors, an error-correcting code contains at most $O(epsilo{n}^{-3/2})$ codewords.

Full work available at URL: https://arxiv.org/abs/2010.16362

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