AI 中文总结
该研究确定了一类q元粘性插入信道的容量，证明满足特定条件的重复信道的香农容量等于零错误容量，并给出了相关显式重复律，这是首次精确确定非平凡重复信道的香农容量。
AI 中文摘要
我们确定了一类q元粘性插入信道的容量。固定q≥2和d≥1，设λ是λ^d=(q−1)(λ^{d−1}+⋯+λ+1)的唯一正解。我们证明，对于所有支撑在1+dℤ≥0上且满足系数支配准则（支配常数γ≥λ^{−d}）的重复律，其香农容量等于零错误容量，二者均为每符号log₂λ比特。我们还给出了满足这些条件的显式重复律，其中一个由加权Fuss–Catalan数给出。据我们所知，这是首次确定非平凡重复信道的香农容量的精确情况。
英文摘要
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.
