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The Fractality of Polar and Reed-Muller Codes.

Bernhard C Geiger1

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PubMed
Summary
This summary is machine-generated.

Generator matrices for polar codes and Reed-Muller codes share properties with fractals. Analyzing index sets reveals fractal-like structures, offering new insights into error correction code properties.

Keywords:
Reed–Muller codesfractalspolar codesself-similarity

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Area of Science:

  • Coding Theory
  • Information Theory
  • Fractal Geometry

Background:

  • Polar codes and Reed-Muller codes are crucial error correction codes.
  • Their generator matrices are derived from a common structure: the Kronecker product of a binary matrix.
  • Row selection criteria differ: Bhattacharyya parameter for polar codes and Hamming weight for Reed-Muller codes.

Purpose of the Study:

  • To investigate the asymptotic properties of index sets used to generate submatrices for polar and Reed-Muller codes.
  • To analyze the fractal characteristics of these index sets as code length increases.
  • To compute key mathematical measures of these sets, such as Lebesgue measure and Hausdorff dimension.

Main Methods:

  • Analysis of generator matrix structures for polar and Reed-Muller codes.
  • Asymptotic analysis of row index sets as blocklength approaches infinity.
  • Computation of Lebesgue measure and Hausdorff dimension for these index sets.
  • Investigation of self-similarity and fine structure properties.

Main Results:

  • The index sets exhibit fractal properties, including self-similarity and fine structure.
  • Specific computations for Lebesgue measure and Hausdorff dimension of these sets were performed.
  • A unified perspective on the mathematical underpinnings of these code families was established.

Conclusions:

  • The study reveals deep connections between coding theory and fractal geometry.
  • The fractal nature of index sets provides a novel framework for understanding error correction codes.
  • These findings can potentially inform the design and analysis of future coding schemes.