The Capacity of a Family of Sticky Channels

Abstract

We determine the capacity of a family of $q$-ary sticky-insertion channels. Fix $q\geq2$ and $d\geq1$, and let $\lambda$ be the unique positive solution of $\lambda^d = (q-1) (\lambda^{d-1} + \cdots + \lambda + 1 )$. We prove that, for every repetition law supported on $1+d\mathbb{Z}_{\geq0}$ and satisfying a coefficientwise-domination criterion with domination constant $\gamma\geq\lambda^{-d}$, the Shannon capacity equals the zero-error capacity, both being $\log_2\lambda$ bits per symbol. We also exhibit explicit repetition laws satisfying these conditions, one of which is given by the weighted Fuss--Catalan numbers. To the best of our knowledge, these are the first known cases of nontrivial repeat channels whose Shannon capacity has been determined exactly.