The {\em binary deletion channel} with deletion probability $d$ ($\text{BDC}_d$) is a random channel that deletes each bit of the input message i.i.d with probability $d$. It has been studied extensively as a canonical example of a channel with synchronization errors. Perhaps the most important question regarding the BDC is determining its capacity. Mitzenmacher and Drinea (ITIT 2006) and Kirsch and Drinea (ITIT 2009) show a method by which distributions on run lengths can be converted to codes for the BDC, yielding a lower bound of $\mathcal{C}(\text{BDC}_d) > 0.1185 \cdot (1-d)$. Fertonani and Duman (ITIT 2010), Dalai (ISIT 2011) and Rahmati and Duman (ITIT 2014) use computer aided analyses based on the Blahut-Arimoto algorithm to prove an upper bound of $\mathcal{C}(\text{BDC}_d) < 0.4143\cdot(1-d)$ in the high deletion probability regime ($d > 0.65$). In this paper, we show that the Blahut-Arimoto algorithm can be implemented with a lower space complexity, allowing us to extend the upper bound analyses, and prove an upper bound of $\mathcal{C}(\text{BDC}_d) < 0.3745 \cdot(1-d)$ for all $d \geq 0.68$. Furthermore, we show that an extension of the Blahut-Arimoto algorithm can also be used to select better run length distributions for Mitzenmacher and Drinea's construction, yielding a lower bound of $\mathcal{C}(\text{BDC}_d) > 0.1221 \cdot (1 - d)$.

翻译：删除概率为$d$的二进制删除信道（$\text{BDC}_d$）是一种随机信道，它以概率$d$独立同分布地删除输入消息的每个比特。作为具有同步错误的信道的典型示例，该信道已被广泛研究。关于BDC最重要的问题或许是确定其容量。Mitzenmacher和Drinea（ITIT 2006）以及Kirsch和Drinea（ITIT 2009）提出了一种方法，通过将游程长度分布转化为BDC的编码，得到下界$\mathcal{C}(\text{BDC}_d) > 0.1185 \cdot (1-d)$。Fertonani和Duman（ITIT 2010）、Dalai（ISIT 2011）以及Rahmati和Duman（ITIT 2014）基于Blahut-Arimoto算法，通过计算机辅助分析证明了在高删除概率区域（$d > 0.65$）上界为$\mathcal{C}(\text{BDC}_d) < 0.4143\cdot(1-d)$。本文证明，Blahut-Arimoto算法可以以更低的空间复杂度实现，从而将上界分析扩展至所有$d \geq 0.68$，并得到上界$\mathcal{C}(\text{BDC}_d) < 0.3745 \cdot(1-d)$。此外，我们表明Blahut-Arimoto算法的扩展还可用于为Mitzenmacher和Drinea的构造选择更优的游程长度分布，从而得到下界$\mathcal{C}(\text{BDC}_d) > 0.1221 \cdot (1 - d)$。