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Error-Correcting Codes and Phase Transitions

Manin, Yuri I. and Marcolli, Matilde (2011) Error-Correcting Codes and Phase Transitions. Mathematics in Computer Science, 5 (2). pp. 133-170. ISSN 1661-8270. https://resolver.caltech.edu/CaltechAUTHORS:20170712-080808371

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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.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1007/s11786-010-0031-8DOIArticle
https://link.springer.com/article/10.1007/s11786-010-0031-8PublisherArticle
https://arxiv.org/abs/0910.5135arXivDiscussion Paper
http://rdcu.be/t9koPublisherFree ReadCube access
Additional Information:© 2010 Birkhäuser Springer. Received: 24 November 2009. Accepted: 17 February 2010. Published online: 23 March 2010.
Issue or Number:2
Record Number:CaltechAUTHORS:20170712-080808371
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170712-080808371
Official Citation:Manin, Y.I. & Marcolli, M. Math.Comput.Sci. (2011) 5: 133. doi:10.1007/s11786-010-0031-8
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78983
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:12 Jul 2017 17:24
Last Modified:03 Oct 2019 18:14

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