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首页> 外文期刊>IEEE Transactions on Information Theory >Optimal Coding for the Binary Deletion Channel With Small Deletion Probability
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Optimal Coding for the Binary Deletion Channel With Small Deletion Probability

机译：删除概率较小的二进制删除通道的最优编码

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摘要

The binary deletion channel is the simplest point-to-point communication channel that models lack of synchronization. Input bits are deleted independently with probability $d$, and when they are not deleted, they are not affected by the channel. Despite significant effort, little is known about the capacity of this channel and even less about optimal coding schemes. In this paper, we develop a new systematic approach to this problem, by demonstrating that capacity can be computed in a series expansion for small deletion probability. We compute three leading terms of this expansion, and find an input distribution that achieves capacity up to this order. This constitutes the first optimal random coding result for the deletion channel. The key idea employed is the following: We understand perfectly the deletion channel with deletion probability $d=0$. It has capacity 1 and the optimal input distribution is iid Bernoulli$(1/2)$ . It is natural to expect that the channel with small deletion probabilities has a capacity that varies smoothly with $d$ , and that the optimal input distribution is obtained by smoothly perturbing the iid Bernoulli$(1/2)$ process. Our results show that this is indeed the case.
机译：二进制删除通道是建模缺乏同步的最简单的点对点通信通道。输入位以概率 $ d $ 独立地删除，并且当不删除它们时，它们不受通道的影响。尽管付出了巨大的努力，但对该信道的容量知之甚少，而关于最佳编码方案的知之甚少。在本文中，我们通过证明可以以较小的删除概率通过级数展开来计算容量，从而针对该问题开发了一种新的系统方法。我们计算了此扩展的三个主导项，并找到了一个可以达到此订单容量的输入分布。这构成了删除信道的第一最佳随机编码结果。所采用的关键思想如下：我们完全理解具有删除概率的删除通道 $ d = 0 $ 。它的容量为1，最佳输入分布为iid Bernoulli $（1/2）$ 。很自然地希望删除概率较小的通道具有随 $ d $ 平滑变化的容量，并且通过平稳地扰动iid Bernoulli $（1/2）$ 过程来获得最佳输入分布。我们的结果表明确实如此。

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