Channels with synchronization errors, exhibiting deletion and insertion errors, find practical applications in DNA storage, data reconstruction, and various other domains. Presence of insertions and deletions render the channel with memory, complicating capacity analysis. For instance, despite the formulation of an independent and identically distributed (i.i.d.) deletion channel more than fifty years ago, and proof that the channel is information stable, hence its Shannon capacity exists, calculation of the capacity remained elusive. However, a relatively recent result establishes the capacity of the deletion channel in the asymptotic regime of small deletion probabilities by computing the dominant terms of the capacity expansion. This paper extends that result to binary insertion channels, determining the dominant terms of the channel capacity for small insertion probabilities and establishing capacity in this asymptotic regime. Specifically, we consider two i.i.d. insertion channel models: insertion channel with possible random bit insertions after every transmitted bit and the Gallager insertion model, for which a bit is replaced by two random bits with a certain probability. To prove our results, we build on methods used for the deletion channel, employing Bernoulli(1/2) inputs for achievability and coupling this with a converse using stationary and ergodic processes as inputs, and show that the channel capacity differs only in the higher order terms from the achievable rates with i.i.d. inputs. The results, for instance, show that the capacity of the random insertion channel is higher than that of the Gallager insertion channel, and quantifies the difference in the asymptotic regime.

翻译：具有同步误差的信道，表现为删除和插入误差，在DNA存储、数据重构及其他多个领域具有实际应用。插入与删除的存在使信道具有记忆性，从而增加了容量分析的复杂性。例如，尽管独立同分布删除信道的模型在五十多年前就已提出，且已证明该信道具有信息稳定性，因而其香农容量存在，但容量的计算一直难以实现。然而，近期的一项成果通过计算容量展开的主项，确立了在删除概率较小的渐近区间内删除信道的容量。本文将该结果推广至二进制插入信道，确定了在小插入概率下信道容量的主项，并建立了该渐近区间内的容量。具体而言，我们考虑两种独立同分布插入信道模型：一种是在每个传输比特后可能发生随机比特插入的插入信道，另一种是Gallager插入模型，其中比特以一定概率被两个随机比特取代。为证明我们的结果，我们基于用于删除信道的方法，采用Bernoulli(1/2)输入证明可达性，并结合使用平稳遍历过程作为输入的逆定理，证明信道容量与独立同分布输入可达速率仅在高阶项上存在差异。例如，结果表明随机插入信道的容量高于Gallager插入信道，并在渐近区间内量化了该差异。