Binomial Channel: On the Capacity-Achieving Distribution and Bounds on the Capacity

Abstract

This work considers a binomial noise channel. The paper can be roughly divided into two parts. The first part is concerned with the properties of the capacity-achieving distribution. In particular, for the binomial channel, it is not known if the capacity-achieving distribution is unique since the output space is finite (i.e., supported on integers $0, \ldots, n)$ and the input space is infinite (i.e., supported on the interval $[0,1]$), and there are multiple distributions that induce the same output distribution. This paper shows that the capacity-achieving distribution is unique by appealing to the total positivity property of the binomial kernel. In addition, we provide upper and lower bounds on the cardinality of the support of the capacity-achieving distribution. Specifically, an upper bound of order $ \frac{n}{2}$ is shown, which improves on the previous upper bound of order $n$ due to Witsenhausen. Moreover, a lower bound of order $\sqrt{n}$ is shown. Finally, additional information about the locations and probability values of the support points is established. The second part of the paper focuses on deriving upper and lower bounds on capacity. In particular, firm bounds are established for all $n$ that show that the capacity scales as $\frac{1}{2} \log(n)$.