The channel output entropy of a transmitted word is the entropy of the possible channel outputs and similarly, the input entropy of a received word is the entropy of all possible transmitted words. The goal of this work is to study these entropy values for the k-deletion, k-insertion channel, where exactly k symbols are deleted, and inserted in the transmitted word, respectively. If all possible words are transmitted with the same probability then studying the input and output entropies is equivalent. For both the 1-insertion and 1-deletion channels, it is proved that among all words with a fixed number of runs, the input entropy is minimized for words with a skewed distribution of their run lengths and it is maximized for words with a balanced distribution of their run lengths. Among our results, we establish a conjecture by Atashpendar et al. which claims that for the binary 1-deletion, the input entropy is maximized for the alternating words. This conjecture is also verified for the 2-deletion channel, where it is proved that constant words with a single run minimize the input entropy.

翻译：传送单词的输出导体是可能的频道输出输出的导体, 类似地, 收到的单词输入的导体是所有可能的传输单词的导体。 这项工作的目标是研究 kdeletion 、 k- 插入通道的这些导体值, 其中精确的 k 符号被删除, 并插入传输单词中。 如果所有可能的单词都以同样的可能性传输, 那么研究输入和输出的异种等同。 对于 1 插入 和 1 列表 频道来说, 已经证明在所有有固定运行数的单词中, 输入的导体最小化为单词, 其运行长度分布偏斜, 且其运行长度分布均衡的单词。 在我们的结果中, 我们建立由 Atashpendar etal 等人 提供的直方字, 其中称对于二进子 1 和输出的导体, 输入的导体是最大化的。 对于交替词, 此导体的导体也被验证为 2 deltion 频道, 其输入的常数以单行最小化的输入 。