We consider binary input deletion/substitution channels, which model certain types of synchronization errors encountered in practice. Specifically, we focus on the regime of small deletion and substitution probabilities, and by extending an approach developed for the deletion-only channel, we obtain an asymptotic characterization of the channel capacity for independent and identically distributed (i.i.d.) deletion/substitution channels. To do so, given a target probability of successful decoding, we first develop an upper bound on the codebook size for arbitrary but fixed numbers of deletions and substitutions, and then extend the result to the case of random deletions and substitutions to obtain a bound on the channel capacity. Our final result is: The i.i.d. deletion/substitution channel capacity is approximately \(1 - H(p_d) - H(p_s)\), for \(p_d, p_s \approx0\), where \(p_d\) and \(p_s\) are the deletion and substitution probabilities, respectively.

翻译：我们研究二进制输入删除/替换信道，该模型刻画了实践中遇到的特定类型同步误差。具体而言，我们关注小概率删除与替换的参数区间，通过扩展针对纯删除信道开发的分析方法，获得了独立同分布删除/替换信道容量的渐近刻画。为实现这一目标，在给定目标译码成功概率的条件下，我们首先建立了任意固定删除与替换数量下的码本规模上界，随后将结果推广至随机删除与替换情形以获得信道容量上界。我们的最终结论表明：当删除概率\(p_d\)与替换概率\(p_s\)趋近于零时，独立同分布删除/替换信道的容量近似为\(1 - H(p_d) - H(p_s)\)，其中\(p_d\)和\(p_s\)分别表示删除概率与替换概率。