The Input and Output Entropies of the $k$-Deletion/Insertion Channel

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, 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 channel, the input entropy is maximized for the alternating words.