2009/10/27 by Yuri I. Manin, Matilde Marcolli, Manin, Yuri I. +1
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #Noncommutative and Quantum Gravity Theories #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.0910.5135
openalex publication_date 2009/10/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.