# Fundamental Limits of Universal Variable-to-Fixed Length Coding of Parametric Sources

## TL;DR

This paper investigates the fundamental limits of universal variable-to-fixed length coding for parametric sources, proposing a scheme based on quantized types and establishing its asymptotic optimality up to the third-order rate.

## Contribution

It introduces a novel coding scheme using quantized type classes and derives its asymptotic third-order rate, proving its optimality among universal codes.

## Key findings

- Third-order coding rate is $H\frac{d}{2}\frac{\log\log M}{\log M}$.
- Proposed scheme achieves this rate asymptotically.
- The rate is proven to be optimal up to the third-order term.

## Abstract

Universal variable-to-fixed (V-F) length coding of $d$-dimensional exponential family of distributions is considered. We propose an achievable scheme consisting of a dictionary, used to parse the source output stream, making use of the previously-introduced notion of quantized types. The quantized type class of a sequence is based on partitioning the space of minimal sufficient statistics into cuboids. Our proposed dictionary consists of sequences in the boundaries of transition from low to high quantized type class size. We derive the asymptotics of the $\epsilon$-coding rate of our coding scheme for large enough dictionaries. In particular, we show that the third-order coding rate of our scheme is $H\frac{d}{2}\frac{\log\log M}{\log M}$, where $H$ is the entropy of the source and $M$ is the dictionary size. We further provide a converse, showing that this rate is optimal up to the third-order term.