An Improved Lower Bound on Support Size of Capacity-Achieving Inputs for the Binomial Channel: Extended version
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2026
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| _version_ | 1866916005567004672 |
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| author | Baniasadi, Mohammadamin Barletta, Luca Dytso, Alex |
| author_facet | Baniasadi, Mohammadamin Barletta, Luca Dytso, Alex |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_12472 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Improved Lower Bound on Support Size of Capacity-Achieving Inputs for the Binomial Channel: Extended version Baniasadi, Mohammadamin Barletta, Luca Dytso, Alex Information Theory 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. |
| title | An Improved Lower Bound on Support Size of Capacity-Achieving Inputs for the Binomial Channel: Extended version |
| topic | Information Theory |
| url | https://arxiv.org/abs/2605.12472 |