Lagrangians, Renormalization, and Quantization in Prefix Coding

TL;DR

This paper introduces a physics-inspired framework for prefix coding using variational principles, renormalization, and quantization, unifying discrete and continuous source coding with entropy bounds and fixed-point structures.

Contribution

It develops a Lagrangian and renormalization approach to prefix coding, including fixed points with iterated-log structures and quantization of continuous laws, extending universal coding theory.

Findings

Fixed points exhibit iterated-log structure

Quantization of continuous laws yields countable prefix codes

Provides entropy bounds and physical analogies for universal coding

Abstract

We develop a statistical mechanics framework for prefix coding based on variational principles, renormalization, and quantization. A Lagrangian formulation of entropy-optimal encoding under the Kraft-McMillan constraint yields a Gibbs-type implied distribution and completeness of the optimal code. A renormalization operator acting on codeword distribution laws produces a coarse-graining flow whose fixed points have iterated-log structure; discrete quantizations of these fixed points include Elias' $\omega$ code as a special case. Extending the theory to mixed discrete-continuous source laws, we show how continuous codelength functions can be quantized into countable prefix codes and derive resolution-adjusted entropy bounds together with Heisenberg-type and Boltzmann-type relations. This provides a unified and physically motivated view of universal coding, with Elias' $\omega$ code as a guiding example.