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

## TL;DR

This paper characterizes all probability mass functions that maximize the minimum expected length of prefix codes for discrete memoryless sources, extending known results beyond the uniform distribution.

## Contribution

It provides a complete characterization of all PMFs where the minimum expected length function attains its maximum, generalizing previous partial results.

## Key findings

- Identifies all PMFs maximizing the minimum expected length.
- Extends known results beyond uniform distributions for specific alphabet sizes.
- Provides theoretical insights into prefix code length optimization.

## Abstract

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.