An Improved Lower Bound on Support Size of Capacity-Achieving Inputs for the Binomial Channel: Extended version

2026-05-12

AI Summary

This study investigates a tight lower bound on the support size of input distributions that achieve the capacity of the binomial channel. By analyzing the structure of the output distribution and leveraging the asymptotic optimality of the Beta-binomial distribution, the authors establish a refined approximation linking channel capacity to the Beta-binomial law, employing tools from information-theoretic capacity analysis, relative entropy, and χ²-divergence comparisons. The main contribution is an improvement of the known lower bound on the support size from √n to the order of √(n log log n), proving that any capacity-achieving input distribution must contain at least this many mass points. Additionally, the paper provides an asymptotic expression for the channel capacity: C(n) = ½ log(nπ/2e) + o(1).

Abstract

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.

Problem

• binomial channel
• capacity-achieving input
• support size
• lower bound
• information theory

Innovation

• binomial channel
• capacity-achieving input
• support size lower bound
• Beta-binomial distribution
• chi-squared divergence