Decoding for Optimal Expected Normalized Distance over the t-Deletion Channel

This paper studies optimal decoding for a special case of the deletion channel, referred by the t-deletion channel, which deletes exactly $t$ symbols of the transmitted word uniformly at random. The goal of the paper is to understand how such an optimal decoder operates in order to minimize the expected normalized distance. A full characterization of a decoder for this setup is given for a channel that deletes one or two symbols. For $t$ = 1 it is shown that when the code is the entire space, the decoder is the lazy decoder which simply returns the channel output. Similarly, for $t$ = 2 it is shown that the decoder acts as the lazy decoder in almost all cases and when the longest run is significantly long, it prolongs the longest run by one symbol.