🤖 AI Summary
This study addresses the Shannon capacity of a class of q-ary sticky insertion channels, wherein symbols are replicated multiple times according to a specific probabilistic rule. By introducing a coefficient dominance condition and leveraging tools from information-theoretic capacity analysis, generating functions, and combinatorics, the authors provide the first exact characterization of the Shannon capacity for this nontrivial replication channel and prove its equivalence to the zero-error capacity. The key contribution lies in constructing an explicit replication law—based on weighted Fuss–Catalan numbers—that satisfies the required conditions, yielding a channel capacity of $\log_2 \lambda$ bits per symbol, where $\lambda$ is uniquely determined by the equation $\lambda^d = (q-1)(\lambda^{d-1} + \cdots + \lambda + 1)$.
📝 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.