Scholarlink

<p>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’ <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\omega\)</EquationSource> </InlineEquation> 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’ <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\omega\)</EquationSource> </InlineEquation> code as a guiding example.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Lagrangians, Renormalization, and Quantization in Prefix Coding

  • 期刊论文

  • 发表日期：

    2026年1月15日

查看全文
  • Alexander Kolpakov,
  • Aidan Rocke

摘要

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.