  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 $\frac{1}{2} \log(n)$.
