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Message transmission in the presence of noise. Second asymptotic theorem and its various formulations

  • Roman V. Belavkin
  • Panos M. Pardalos
  • Jose C. Principe
Chapter

Abstract

In this chapter, we provide the most significant asymptotic results concerning the existence of optimal codes for noisy channels. It is proven that the Shannon’s amount of information is a bound on Hartley’s amount of information transmitted with asymptotic zero probability of error. This is the meaning of the second asymptotic theorem. Further we provide formulae showing how quickly the probability of error for decoding decreases when the block length increases. Contrary to the conventional approach, we represent the above results not in terms of channel capacity (i.e., we do not perform the maximization of the limit amount of information with respect to the probability density of the input variable), but in terms of Shannon’s amount of information.

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Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  • Roman V. Belavkin
    • 1
  • Panos M. Pardalos
    • 2
  • Jose C. Principe
    • 3
  1. 1.Faculty of Science and TechnologyMiddlesex UniversityLondonUK
  2. 2.Industrial and Systems EngineeringUniversity of FloridaGainesvilleUSA
  3. 3.Electrical & Computer EngineeringUniversity of FloridaGainesvilleUSA

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