We consider binary error correcting codes when errors are deletions. A basic challenge concerning deletion codes is determining ${p}_{0}^{(adv)}$ , the zero-rate threshold of adversarial deletions, defined to be the supremum of all $p$ for which there exists a code family with rate bounded away from 0 capable of correcting a fraction $p$ of adversarial deletions. A recent construction of deletion-correcting codes shows that ${p}_{0}^{(adv)} \ge \sqrt {2}-1$ , and the trivial upper bound, ${p}_{0}^{(adv)}\le {1}/{2}$ , is the best known. Perhaps surprisingly, we do not know whether or not ${p}_{0}^{(adv)} = 1/2$ . In this work, to gain further insight into deletion codes, we explore two related error models: oblivious deletions and online deletions, which are in between random and adversarial deletions in power. In the oblivious model, the channel can inflict an arbitrary pattern of ${pn}$ deletions, picked without knowledge of the codeword. We prove the existence of binary codes of positive rate that can correct any fraction ${p} < 1$ of oblivious deletions, establishing that the associated zero-rate threshold ${p}_{0}^{(obliv)}$ equals 1. For online deletions, where the channel decides whether to delete bit ${x}_{{i}}$ based only on knowledge of bits ${x}_{1}{x}_{2} {\dots }{x}_{{i}}$ , define the deterministic zero-rate threshold for online deletions ${p}_{0}^{(on,{d})}$ to be the supremum of $p$ for which there exist deterministic codes against an online channel causing ${pn}$ deletions with low average probability of error. That is, the probability that a randomly chosen codeword is decoded incorrectly is small. We prove ${p}_{0}^{(adv)}={1}/{2}$ if and only if ${p}_{0}^{(on,{d})}={1}/{2}$ .

中文翻译：

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针对不经意和在线模型中的删除进行编码

当错误是删除时，我们考虑二进制纠错码。关于删除代码的一个基本挑战是确定 ${p}_{0}^{(adv)}$ ， 这 对抗性删除的零率阈值，定义为所有 $p$ 存在一个码族，其速率从 0 开始，能够纠正一个分数 $p$ 的对抗性删除。最近的删除纠正代码的构建表明， ${p}_{0}^{(adv)} \ge \sqrt {2}-1$ ，以及平凡的上限， ${p}_{0}^{(adv)}\le {1}/{2}$ ，最为人所知。也许令人惊讶的是，我们不知道是否 ${p}_{0}^{(adv)} = 1/2$ . 在这项工作中，为了进一步了解删除代码，我们探索了两种相关的错误模型：不经意删除和在线删除，它们介于随机删除和对抗性删除之间。在遗忘模型中，通道可以造成随意的 的模式 ${pn}$ 删除，在不知道码字的情况下挑选。我们证明了可以纠正任何分数的正率二进制代码的存在 ${p} < 1$ 不经意删除，建立相关的零率阈值 ${p}_{0}^{(obliv)}$ 等于1。对于在线删除，由通道决定是否删除位 ${x}_{{i}}$ 仅基于位的知识 ${x}_{1}{x}_{2} {\dots}{x}_{{i}}$ ，定义 在线删除的确定性零率阈值 ${p}_{0}^{(on,{d})}$ 成为至高无上的 $p$ 存在针对在线渠道的确定性代码导致 ${pn}$ 删除率低 平均数错误的概率。也就是说，随机选择的码字被错误解码的概率很小。我们证明 ${p}_{0}^{(adv)}={1}/{2}$ 当且仅当 ${p}_{0}^{(on,{d})}={1}/{2}$ .