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This study explores how noise affects data codes, aiming for uniform distributions using Rényi divergence. It identifies specific code families that achieve this "perfect smoothing" for secure communication and error correction.

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Area of Science:

  • Information Theory
  • Coding Theory
  • Harmonic Analysis

Background:

  • Noise operators transform codes into distributions.
  • Channel resolvability is key for security and error correction.
  • Uniform distributions are a benchmark for noise resilience.

Purpose of the Study:

  • Characterize conditions for output distributions to approach uniformity under noise.
  • Quantify requirements for asymptotic uniformity (perfect smoothing).
  • Identify explicit code families achieving perfect smoothing.

Main Methods:

  • Utilizing Rényi divergence of order α∈(1,∞] to measure closeness to uniform distribution.
  • Applying recent advances in harmonic analysis of functions on the Hamming space.
  • Analyzing Wyner's transmission scheme for the binary wiretap channel.

Main Results:

  • Derived expressions for minimum code rates for asymptotic perfect smoothing.
  • Identified explicit code families (e.g., nested Reed-Muller codes) achieving perfect smoothing.
  • Demonstrated strong secrecy guarantees in Wyner's scheme using these codes.

Conclusions:

  • Nested Reed-Muller codes enable reliable and secure communication over binary symmetric wiretap channels.
  • Established a direct link between the concepts of smoothing and error correction.
  • The findings have implications for cryptography, data compression, and complexity theory.