Probability Mass Functions for which Sources have the Maximum Minimum Expected Length

Let ${\mathcal{P}}_n$ be the set of all probability mass functions (PMFs) $(p_1,p_2,\dots ,p_n)$ that satisfy $p_i>0$ for $1\le i\le n$. Define the minimum expected length function $$\mathcal{L}$_D :$\mathcal{P}$_n \rightarrow $\mathbb{R}$$suchthat$$\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.