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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 theminimum expected length function $\mathcal{L}_D :\mathcal{P}_n \rightarrow\mathbb{R}$ such that $\mathcal{L}_D (P)$ is the minimum expected length of aprefix code, formed out of an alphabet of size $D$, for the discrete memorylesssource having $P$ as its source distribution. It is well-known that thefunction $\mathcal{L}_D$ attains its maximum value at the uniform distribution.Further, when $n$ is of the form $D^m$, with $m$ being a positive integer, PMFsother than the uniform distribution at which $\mathcal{L}_D$ attains itsmaximum value are known. However, a complete characterization of all such PMFsat which the minimum expected length function attains its maximum value has notbeen done so far. This is done in this paper.

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