We study the binomial channel and the structure of its capacity-achieving input and output distributions. It is known that the capacity-achieving input distribution is discrete and supported on finitely many points. The best previously known bounds show that the support size of the capacity-achieving distribution is lower-bounded by a term of order $\sqrt n$ and upper-bounded by a term of order $n/2$, where $n$ is the number of trials. In this work, we derive a new lower bound on the support size of order $\sqrt{n\log\log n}$, up to explicit constants. The proof consists of three main steps. First, we derive new upper and lower bounds on the capacity with a gap that vanishes as $n\to\infty$, which yields $C(n)=\frac12\log\frac{nπ}{2e}+o(1)$. Second, we show that the Beta-binomial output distribution induced by the reference input $X_r\sim\mathrm{Beta}(1/2,1/2)$ is asymptotically optimal: it approaches the capacity-achieving output distribution in relative entropy and, after a comparison step, in $χ^2$ divergence. Third, we prove a quantitative $χ^2$ approximation lower bound showing that this Beta-binomial output cannot be approximated too well by the output induced by a $K$-point input. Combining these ingredients forces the capacity-achieving input distribution to have at least order $\sqrt{n\log\log n}$ mass points.

翻译：我们研究二项信道及其容量可达输入与输出分布的结构。已知容量可达输入分布是离散的，且支撑于有限个点上。此前最优的下界和上界表明，容量可达分布的支撑集大小分别受限于$\sqrt n$量级和$n/2$量级，其中$n$为试验次数。本文推导了支撑集大小的新下界，其量级为$\sqrt{n\log\log n}$（含显式常数）。证明分为三个主要步骤：首先，我们推导容量新的上界与下界，其差距随$n\to\infty$趋于零，得到$C(n)=\frac12\log\frac{nπ}{2e}+o(1)$；其次，我们证明由参考输入$X_r\sim\mathrm{Beta}(1/2,1/2)$诱导的Beta-二项输出分布是渐近最优的：该分布在相对熵意义下逼近容量可达输出分布，且经对比步骤后在$\chi^2$散度下同样逼近；最后，我们证明一个定量的$\chi^2$逼近下界，表明该Beta-二项输出不能被任何$K$点输入诱导的输出分布过好地逼近。综合这些要素，迫使容量可达输入分布至少具有$\sqrt{n\log\log n}$量级的质量点。