Maximal input entropy for binary two-deletion and two-insertion channels

Abstract

Under uniform transmission, we determine the received binary words that maximize input entropy for two-deletion and two-insertion channels. For two deletions the maximizers are exactly the two alternating words at every positive received length. For two insertions the same conclusion holds from received length five onward; we give all smaller cases and exact maximal values. The proof combines an embedding-count identity organized by runs with strict convexity estimates, and a grouping of two-position deletion masks. For every insertion radius we give the complete positive embedding histogram of an alternating received word. We also determine its three-deletion histogram for every length, and certify all three-deletion maximizers through received length twelve and all three-insertion maximizers through length eighteen by exact integer comparison. These computations give precise three-error conjectures with their small-length exceptions.