  Let $\mathcal{P}_n$ be the set of all probability mass functions (PMFs)
$(p_1,p_2,\ldots,p_n)$ that satisfy $p_i>0$ for $1\leq i \leq n$. Define the
minimum expected length function $\mathcal{L}_D :\mathcal{P}_n \rightarrow
\mathbb{R}$ such that $\mathcal{L}_D (P)$ is the minimum expected length of a
prefix code, formed out of an alphabet of size $D$, for the discrete memoryless
source having $P$ as its source distribution. It is well-known that the
function $\mathcal{L}_D$ attains its maximum value at the uniform distribution.
Further, when $n$ is of the form $D^m$, with $m$ being a positive integer, PMFs
other than the uniform distribution at which $\mathcal{L}_D$ attains its
maximum value are known. However, a complete characterization of all such PMFs
at which the minimum expected length function attains its maximum value has not
been done so far. This is done in this paper.
