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The channel output entropy of a transmitted word is the entropy of thepossible channel outputs and similarly, the input entropy of a received word isthe entropy of all possible transmitted words. The goal of this work is tostudy these entropy values for the k-deletion, k-insertion channel, whereexactly k symbols are deleted, and inserted in the transmitted word,respectively. If all possible words are transmitted with the same probabilitythen studying the input and output entropies is equivalent. For both the1-insertion and 1-deletion channels, it is proved that among all words with afixed number of runs, the input entropy is minimized for words with a skeweddistribution of their run lengths and it is maximized for words with a balanceddistribution of their run lengths. Among our results, we establish a conjectureby Atashpendar et al. which claims that for the binary 1-deletion, the inputentropy is maximized for the alternating words. This conjecture is alsoverified for the 2-deletion channel, where it is proved that constant wordswith a single run minimize the input entropy.

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