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) $(p_{1},p_{2},\ \ldots, p_{n})$ 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.