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Statistical mechanics of error exponents for error-correcting codes.

Thierry Mora1, Olivier Rivoire

  • 1Laboratoire de Physique Théorique et Modèles Statistiques, Bâtiment 100, Université Paris-Sud, F-91405 Orsay, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|February 7, 2007
PubMed
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We introduce a new thermodynamic formalism to exactly compute error exponents for error-correcting codes. This method reveals two phase transitions related to noise intensity, offering insights into code performance.

Area of Science:

  • Information Theory
  • Statistical Physics
  • Coding Theory

Background:

  • Error exponents quantify the performance of error-correcting codes.
  • Exact computation of error exponents has been a long-standing challenge.

Purpose of the Study:

  • To develop a general formalism for the exact computation of error exponents.
  • To analyze the behavior of error exponents in low-density parity-check codes.

Main Methods:

  • Introduction of a thermodynamic formalism.
  • Application of the cavity method for large deviations.
  • Analysis of maximum-likelihood decoding on binary erasure and symmetric channels.

Main Results:

  • Derivation of expressions for average and typical error exponents.

Related Experiment Videos

  • Identification of two distinct phase transitions as noise intensity decreases.
  • Characterization of transitions as glass-to-ferromagnetic and paramagnetic-to-glass.

    • Conclusions:

    • The proposed thermodynamic formalism provides a powerful tool for analyzing error-correcting codes.
    • The discovered phase transitions offer new perspectives on code behavior under varying noise conditions.