The Capacity of a Family of Sticky Channels
arXiv:2607.28281
The paper exactly determines the Shannon and zero‑error capacity of a family of q‑ary sticky‑insertion (repeat) channels, showing it equals log₂ λ bits per symbol under specific repetition laws.
Abstract
We determine the capacity of a family of $q$-ary sticky-insertion channels. Fix $q\geq2$ and $d\geq1$, and let $λ$ be the unique positive solution of $λ^d = (q-1) (λ^{d-1} + \cdots + λ+ 1 )$. We prove that, for every repetition law supported on $1+d\mathbb{Z}_{\geq0}$ and satisfying a coefficientwise-domination criterion with domination constant $γ\geqλ^{-d}$, the Shannon capacity equals the zero-error capacity, both being $\log_2λ$ 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.