Varieties of codes and kraft inequality

  • Authors:

  • Fabio Burderi;Antonio Restivo

  • Affiliations:

  • Dipartimento di Matematica ed Applicazioni, Università degli studi di Palermo, Palermo, Italy;Dipartimento di Matematica ed Applicazioni, Università degli studi di Palermo, Palermo, Italy

  • Venue:
  • STACS'05 Proceedings of the 22nd annual conference on Theoretical Aspects of Computer Science
  • Year:
  • 2005

  • Preface

    Theoretical Computer Science - The art of theory

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Abstract

Decipherability conditions for codes are investigated by using the approach of Guzmán, who introduced in [7] the notion of variety of codes and established a connection between classes of codes and varieties of monoids. The class of Uniquely Decipherable (UD) codes is a special case of variety of codes, corresponding to the variety of all monoids. It is well known that the Kraft inequality is a necessary condition for UD codes, but it is not sufficient, in the sense that there exist codes that are not UD and that satisfy the Kraft inequality. The main result of the present paper states that, given a variety $\mathcal{V}$ of codes, if all the elements of $\mathcal{V}$ satisfy the Kraft inequality, then $\mathcal{V}$ is the variety of UD codes. Thus, in terms of varieties, Kraft inequality characterizes UD codes.