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Error-Correcting Codes and Phase Transitions

Mathematics in Computer Science volume 5, pages 133–170 (2011)Cite this article

Abstract

The theory of error-correcting codes is concerned with constructing codes that optimize simultaneously transmission rate and relative minimum distance. These conflicting requirements determine an asymptotic bound, which is a continuous curve in the space of parameters. The main goal of this paper is to relate the asymptotic bound to phase diagrams of quantum statistical mechanical systems. We first identify the code parameters with Hausdorff and von Neumann dimensions, by considering fractals consisting of infinite sequences of code words. We then construct operator algebras associated to individual codes. These are Toeplitz algebras with a time evolution for which the KMS state at critical temperature gives the Hausdorff measure on the corresponding fractal. We extend this construction to algebras associated to limit points of codes, with non-uniform multi-fractal measures, and to tensor products over varying parameters.

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Authors and Affiliations

  1. Max Planck Institute for Mathematics, Bonn, Germany

    Yuri I. Manin & Matilde Marcolli

  2. Mathematics Department, Northwestern University, Evanston, IL, USA

    Yuri I. Manin

  3. Department of Mathematics, California Institute of Technology, Pasadena, CA, USA

    Matilde Marcolli

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  1. Yuri I. Manin
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  2. Matilde Marcolli
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Correspondence to Matilde Marcolli.

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Manin, Y.I., Marcolli, M. Error-Correcting Codes and Phase Transitions. Math.Comput.Sci. 5, 133–170 (2011). https://doi.org/10.1007/s11786-010-0031-8

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  • DOI: https://doi.org/10.1007/s11786-010-0031-8

Keywords

  • Partition Function
  • Hausdorff Dimension
  • Hausdorff Measure
  • Kolmogorov Complexity
  • Code Domain